Well there's a clear divide bewteen the creationists and the evolutionists here. If you're a creationist, then you believe that God made a chicken first, which then laid an egg.
But if you believe in evolution (as I certainly do) then it is the egg which comes first. The reason is simple. A bird which could (just) be described as sufficiently chicken-like to be called a chicken, must at some point have been hatched from an egg laid by a bird which was very similar to a chicken, but not quite sufficiently chicken-like, to be called a chicken. Such a change from almost a chicken to a true chicken is a mutational change, and a mutational change can only occur during the formation of a fertilised egg.
Therefore, the first egg which was sufficiently chicken-like to be called a chicken egg, was laid before the first bird to be sufficiently chicken-like to be called a chicken was hatched from the egg.
Therefore, the answer to the question as to which came first, is that the chicken egg came before the chicken.
And I have never written so many chicken references in a single article ever before.