I think there are two valid scientific/philosophical answers without taking religion into it, based on one question:

Are we specifically talking a chicken egg, or the concept of an egg?

In the former case, eggshells contain compounds that cannot exist in nature, and must come from a creature. a chicken egg cannot exist without a chicken before it, thus the chicken came first.

In the latter case, various evolutionary splits happened between animals evolving egg developing capability and some animals evolving into chickens. From this we can say that the egg came before the chicken.

Worst case, this solved exactly nothing. Best case, it can be an exercise in reasoning.