Eggs predate chickens by millions of years, so the first egg came long before the first chicken. Presumably, however, the question is supposed to be something like: “which came first, the first chicken or the first chicken egg?” But to answer this question, we first have to ask something about the eggs themselves: are all chicken eggs laid by chickens, or are all chickens born from chicken eggs? If it’s the former, then the first chicken necessarily came before the first chicken egg. If it’s the latter, the first chicken egg necessarily came before the first chicken.
This was a problem in ancient times because it was assumed that species did not evolve. They might change a bit, but never so much as to become something else entirely. If so, then all chickens had been born from chicken eggs and all chicken eggs had been laid by chickens. The potential paradox this seems to create is the origin of the question, by the way. And while we treat the question as a silly sideshow these days, back then it was a entry point into larger issues of first causes, infinite regresses, and the origins of the universe.
These days, however, we have a different problem. While the theory of evolution eliminates the threats of paradox and infinite regress, it raises the question of how to demarcate species (the species problem). Is the dividing line between one species and another ultimately arbitrary, or is it possible for taxonomists to “carve nature at the joints” (as the old, and rather apropos, saying goes)? In other words, do we decide or discover where one species ends and the next one begins? This is the debate between nominalism and realism.
Realism would give us the promise of a definitive answer, but it is hard to reconcile with our current understanding of evolution. Nominalism means that the answer is whatever we want it to be (within certain limits), but is potentially unsatisfying because there are decent arguments for both “all chicken eggs are laid by chickens” and “all chickens are born from chicken eggs.”
What would you call an egg laid by a chicken, after all, but a chicken egg? And if we say that all eggs laid by chickens are chicken eggs, then all eggs laid by proto-chickens would be proto-chicken eggs (even if they hatch a chicken). Thus the first chicken came before the first chicken egg.
But wait! What if you came across an unidentified egg and then saw a chicken hatch from it? If we decide to call it a chicken egg even in the absence of any knowledge about what sort of creature laid it, then we are saying that a chicken hatching from an egg is enough to make that egg a chicken egg. Thus the first chicken egg came before the first chicken.
There are facts about the domestication of chickens from junglefowl that could play a role in our decision, but I think a comparison to similar cases that lack the complicating factor of the egg might be more helpful. Specifically, it seems to me that the first chicken could come from a non-chicken egg for the same reason that the first human could gestate within a non-human. And so, I lean slightly towards the idea that the first chicken came before the first chicken egg.