Well, if you interpret the question like that, it's clear that eggs have been around much longer than chicken. However, you could also think about what came first, the chicken or the chicken egg. Now, there is no well-defined boundary between a species and the species that evolved into it. So to answer the question, we have to assume that there is indeed one individual that is the first that we can call "chicken". Then its parents were by assumption not chicken, but another species of bird. We can now clearly understand that under these assumptions, the question is a question of semantics: if a chicken egg is an egg that is laid by a chicken, then the chicken came first. If a chicken egg is an egg from which a chicken is hatched, then the chicken egg came first.
Why is this interesting? Because normally, both definitions of "chicken egg" are the same, since chicken lay eggs from which chicken are hatched.