Answer with, "The egg." The chicken is not the first egg-laying animal. There were many other birds and reptiles that came before it, and they layed eggs. So there were clearly eggs before there were chickens.
Your questioner may counter with, "Fine. Which came first, the chicken or the chicken egg?" The proper response to this is, "Define 'chicken egg'." Is a chicken egg and egg that comes from a chicken or is it an egg that produces a chicken. If they come up with a response and actually answer the question, their response directly informs your answer. If they say that a chicken egg comes from a chicken, then the chicken came first. If they say that a chicken egg produces a chicken, then the egg comes first.
I don't really believe the proper response is either though, since those aren't the only two choices. I think we can all say, without doubt, that if an egg both comes from a chicken and produces a chicken then it is a chicken egg. In that case, the chicken must come first.
But what type of egg comes from something that isn't a chicken and yet produces a chicken? That's the real question.