From an evolutionary standpoint, the egg came first, of course. Eggs evolved many millions of years before birds, let alone chickens. Granted, they were the jelly-like fish eggs, and then the leathery, soft shelled reptilian-like eggs, and the hard shelled eggs we associate with modern chickens would be even further down the line, but they were eggs nonetheless.
However, in the Latin-based English alphabet, "chicken" comes first, because C is before E, so you'll find "chicken" before "egg" in the dictionary. But this is not applicable, of course, to all languages - for instance, "huevo" (Spanish for "egg") comes alphabetically before "pollo" (Spanish for "chicken").
And from a strictly single-individual, linear point of view, the egg comes first, because every living thing begins as some form of "fetus," and for chickens, that takes the form of an external egg that must be brooded over by a mother hen - who, of course, came from an egg, laid by her mother hen, who came from an egg, and so on. Thus, the egg is the starting point.
Except, of course, for the fact that the chicken egg cannot exist without the chicken, which was domesticated by ancient humans out of wild birds, so what we call a "chicken egg" today cannot exist without the larger foundation of "chicken" as defined by humans (who just think they know everything, including how to define nebulous concepts). So the chicken had to come first, as long as we're specifically talking about chicken eggs.
Except no one ever specifies that, do they? They imply it, sure, but that just leaves so much open to interpretation - and if we're going to be really granular, no one ever specifies what kind of chicken, either. You're just supposed to imagine a generic, perhaps slightly cartoonish, hen sitting on a nest of eggs. That's pretty imprecise wording for a question that challenges a precise answer.
And if we do get into the specific breeds of chicken, then we look at where those lines started, and it's with other chickens, who were selectively bred to produce eggs that would contain chicks with desired traits.
Don't even get me started on tracing chickens back to dinosaurs, because then we end up back at the first point, which is that the first eggs evolved before bones did, much less chickens, and then we loop back to the fourth point, about defining what a "chicken" is in order to define what a "chicken egg" is.
Honestly, at this point, I'm pretty sure the poor things are crossing the road to get away from the purposefully-vague questions that philosophers keep bothering them with.