It is only natural that, when confronted with timeless and confounding questions, your friends should turn to you, the philosopher. Sooner or later, then, they will ask you which came first, the chicken or the egg. You must be prepared to discuss this issue in pedantic depth or lose your reputation for intimidating scholarly acumen. Only after understanding this issue will you be prepared for even deeper and more troubling questions such as "Is water wet? Or is water only something that makes other things wet?"
The question invites us to consider a sequence of the following sort, stretching back in time: chicken, egg, chicken, egg, chicken.... The first term of the series can be chosen arbitrarily. The question is the terminus. If one assumes an infinite past and everlasting species, there may be no terminus. However, the cosmological assumptions behind such a view are highly doubtful. Therefore, it seems, there must be a terminus member of the series, temporally first, either a chicken or an egg. The question which came first is often posed rhetorically as though it were obvious that there could be no good epistemic grounds for choice. However, as I aim to show, this appearance of irresolvability is misleading. The egg came first.
Young Earth Creationist views merit brief treatment. If God created chickens on the Fourth Day along with "every kind of winged creature", then the question is whether He chose to create the chicken first, the egg first, both types simultaneously, or a being at the very instant of transition between egg and chicken (when it is arguably either both or neither). The question thus dissolves into the general mystery of God's will. Textual evidence somewhat favors either the chicken or both, since God said "let birds fly above the earth" and the Bible then immediately states "and so it was", before transition to the Fifth Day. So at least some winged creatures were already flying on the Fourth Day, and one day is ordinarily insufficient time for eggs to mature into flying birds. Since chickens aren't much prone to fly, though, it's dubious whether such observations extend to them, unless God implemented a regular rule in which winged creatures were created either mature or simultaneously in a mix of mature and immature states. And in any case, it is granted on all sides that events were unusual and not subject to the normal laws of development during the first Six Days.
If we accept the theory of evolution, as I think we should, then the chicken derives from a lineage that ultimately traces back to non-chickens. (The issues here are the same whether we consider the domestic chicken to be its own species or whether we lump it together with the rest of gallus gallus including the Red Junglefowl from which the domestic chicken appears to be mostly descended.) The first chicken arose either as a hybrid of two non-chickens or via mutation from a non-chicken. Consider the mutation case first. It's improbable (though not impossible) that between any two generations in avian history, X and X-1, there would be enough differentiation for a clean classification of X as a chicken and X-1 as a non-chicken. Thus we appear to have a Sorites case. Just as it seems that adding one grain to a non-heap can't make it a heap, resulting in the paradox that no addition of single grains could ever make a heap, so also one might worry that one generation's difference could never (at least with any realistic likelihood) make the difference between a chicken and a non-chicken, resulting in the paradox of chickens in the primordial soup.
Now there are things philosophers can do about these paradoxes. Somehow heaps arise, despite the argument above. One simple approach is epistemicism, according to which there really is a sharp line in the world such that X-1 is a non-heap and X is a heap, X-1 is a non-chicken and X is a chicken. On this view, our inability to discern this line is merely an epistemic failure on our part. Apparent vagueness is really only ignorance. Another simple approach is to allow that there really are vague properties in the world that defy classification in the two-valued logic of true and false. On this view, between X, which is definitely a chicken, and X-N, which is definitely a non-chicken, there are some vague cases of which it is neither true nor false that it is a chicken, or somehow both true and false, or somewhere between true and false, or something like that. There are also more complicated views, too, than these, but we needn't enter them, because one key point remains the same across all these Sorites approaches: The Sorites cases progress not as follows: X chicken, X-1 egg, X-2 chicken, X-3 egg, X-4 chicken.... Rather, they progress in chicken-egg pairs. From a genetic perspective, since the chicken and egg share DNA, they form a single Sorites unit. Within this unit, the egg clearly comes first, since the chicken is born from the egg, sharing its DNA, and there is a DNA difference between the egg and the hen from which that egg is laid. For a ridiculous argument to the contrary, see here.
If we turn to the possibility of speciation by hybridization, similar considerations apply.
A much poorer argument for the same conclusion runs as follows: Whatever ancestor species gave rise to chickens presumably laid eggs. Therefore, there were eggs long before there were chickens. Therefore, the egg came first. The weakness in this argument is that it misconstrues the original question. The question is not "Which came first, chickens or eggs?" but rather "Which came first, the first chicken or the first chicken egg?"
However, the poverty of this last argument does raise vividly the issue of how one assigns eggs to species. The egg-first conclusion could be evaded if we typed eggs by reference to the mother: If the mother is a chicken, the egg is a chicken egg; if the mother is not a chicken, the egg is not a chicken egg. David Papineau succinctly offers the two relevant considerations against such a view here. First, if we type by DNA, which would seem to be the default biological standard, the egg shares more of its DNA with the hatchling than with its parent. Second, as anyone can see via intuitive armchair reflection on a priori principles: "If a kangaroo laid an egg from which an ostrich hatched, that would surely be an ostrich egg, not a kangaroo egg."
(HT: Walter Sinnott-Armstrong, who in turn credited Roy Sorenson.)
Update, Feb. 2:
In the comments, Papineau reveals that he has recanted in light of considerations advanced by Mohan Matthen in his important but so far sadly neglected "Chicken, Eggs, and Speciation" -- considerations also briefly mentioned by Ron Mallon in his comment. Although I find merit in these remarks, I am not convinced and I believe Papineau has abandoned the egg-first view too precipitously.
Matthen argues that: "Speciation occurs when a population comes to be reproductively isolated because the last individual that formerly bridged that population to others died, or because this individual ceased to be fertile (or when other integrating factors cease to operate)" (2009, p. 110). He suggests that this event will normally occur when both soon-to-be-chickens and soon-to-be-chicken-eggs exist in the population. Thus, he concludes, a whole population of chickens and eggs is simultaneously created in a single instant. In assessing this view let me note first that depending on the size of the population and its egg-laying habits, this view might suggest a likelihood of chickens first. Suppose that in a small population of ancestral pre-chickens the last bridge individual dies outside of laying season; or suppose that the end of an individual's last laying season marks the end of an individual's fertility. If there are no out-of-season eggs at the crucial moment, then chickens came first.
More importantly, however, Matthen's criterion of speciation leads to highly counterintuitive and impractical results. Matthen defines reproductive isolation between populations in terms of the probability of gene transfer between those populations. (Also relevant to his distinction is the shape of the graph of the likelihood of gene transfer by number of generations, but that complication isn't relevant to the present issue.) But probability of gene transfer can be very sharply affected by factors that don't seem to create comparably sizable influences on species boundaries. So, for example, when human beings migrated to North America, the probability of gene transfer with the ancestral population declined sharply, and soon became essentially zero (and in any case in excess of the probability of gene transfer between geographically coincident hybridizing species). By Matthen's criterion, this would be a speciating event. After Columbus, gene transfer probability slowly rose and by now gene transfer is very high between individuals with Native American ancestry and those without. Thus, by Matthen's criterion, Native Americans were for several thousand years a distinct species -- not homo sapiens! -- and now they are homo sapiens again. If the moment of change was Columbus's first landing (or some other discrete moment), then the anchoring of a ship, or some other event, perhaps a romantic interlude between Pocahontas and John Smith, caused everyone on the two continents simultaneously to change species!
More simply, we might imagine a chicken permanently trapped in an inescapable cage. Its probability of exchanging genes with other individuals is now zero. Since Matthen allows for species consisting of a single individual, this chicken has now speciated. Depending on how we interpret the counterfactual probabilities, we might even imagine opening and shutting the door repeatedly (perhaps due to some crazy low-probability event) causing that individual to flash repeatedly back and forth between being a chicken and being a non-chicken, with no differences in morphology, actual behavior, location, or sexual preference during the period. On the surface, it seems that Matthen's criterion might even result in all infertile individuals belonging to singleton species.
There are both philosophical and practical biological reasons not to lightly say that individuals may change species during their lifetimes. One consideration is that of animal identity. If I point at an individual chicken and ask at what point the entity at which I am pointing ceases to exist, there are good practical (and maybe metaphysical) reasons to think that the entity does not cease to exist when a single feather falls off, nor to think that it continues to exist despite being smushed into gravy. The most natural and practical approach, it seems, is to say that the entity to which I intend to refer (in the normal case) is essentially a chicken and thus that it continues to exist exactly as long as it remains a chicken. Consequently, on the assumption that the individual pre-chicken avians don't cease to exist when they become reproductively isolated, they remain non-chickens despite overall changes in the makeup of the avian population. (These individuals may, nonetheless, give birth to chickens.) Nor does it seem that any important scientific biological purpose would be served by requiring the relabeling of individual organisms, depending on population movements, once those organisms are properly classified. Long-enduring organisms, such as trees, seem best classified as members of the ancestral population they were born into, even if their species has moved on since. Long-lived individuals can remain as living remnants of the ancestral species -- a species with temporally ragged but individual-respecting borders. The attractiveness of this view is especially evident if we consider the possibility of thawing a long-frozen dinosaur egg.
Matthen argues as follows against the those who embrace either an egg-first or a chicken-first view: The first chicken would need to have descendants by breeding with a non-chicken, but since by definition species are reproductively isolated this view leads to contradiction. This consequence is easily evaded with the right theory of vagueness and a suitable interpretation of the reproductive isolation criterion. On my preferred theory of vagueness, there will be individuals of which it's neither determinately true nor determinately false that they are chickens. We can then define reproductive isolation as the view that no individual of which it is determinately true that it is a member of species X can reproduce with an individual of which it is determinately false that it is a member of species X. As long as all breeding is between determinate members and individuals in the indeterminate middle, the reproductive isolation criterion is satisfied. (This is not to concede, however, that species should be defined entirely in terms of reproductive isolation, given the problems in articulating that criterion plausibly, some of which are noted above.)
Second update, Feb. 3:
The issues prove even deeper and more convoluted than I thought! In the comments section, Matthen has posted a reply to my objections, which we pursue for a couple more conversational turns. Although I'm not entirely ready to accept his account of species, I see merit in his thought that the best unit of evaluation might be the population rather than the individual, and if there is a first moment at which the population as a whole becomes a chicken population (rather than speciation involving temporally ragged but individual-respecting borders), then that might be a moment at which multiple avians and possibly multiple avian eggs simultaneously become chickens and chicken eggs.
An anonymous reader raises another point that seems worth developing. If we think of "chickens" not exclusively in terms of their membership in a biologically discriminable species but at least partly in terms of their domestication, then the following considerations might favor a chicken-first perspective. Some act of domestication -- either an act of behavioral training or an act of selection among fowl -- was the last-straw change from non-chickenhood to chickenhood, creating the first chicken. But this act was very likely performed on a newly-hatched or adult bird, not on an egg, since eggs are not trainable and hard to discriminate usefully among. Therefore the first entity in the chicken-egg sequence was a chicken, not an egg. For some reason, I find it much more natural to accept the possibility that a non-chicken could become a chicken mid-life if chickenhood is conceived partly in terms of domestication than if it is conceived entirely as a matter of traditional biological species. (I'm not sure how stable this argument is, however, across different accounts of vagueness.)