I have two answers for this.
First, from a stand point of species differentiation, well, it's an extremely slow and gradual process. The 'protochicken' or parent to the first chicken would essentially be the same animal. Same with the chicken egg and protochicken egg. In that way, neither really came first, they eventually just kinda.. happened.
Second, something I picked up in college: the egg has all the makings of a chicken, but no way to synthesize one of the proteins vital in its shell. Meanwhile, in a way, the adult chicken has the ability to make both an egg and a child chicken. Therefore, the egg depends on the chicken to produce it, but the chicken can self reproduce the fetus 'and' the egg. Chicken has to come first in this case.
EDIT: this argument kind of assumes we're talking about a chicken egg, haha. Otherwise, fish laid eggs, and fish predate chickens, thus the egg obviously comes first.