For me, the chicken is just an allegory. The question really is, what came first, the egg, or the animal that produced it? If we accept this scenario, then its obviously the animal.If we go back to the first unicellular and then multicellular organisms, reproduction was purely by mitosis. So the first universal ancestors pre-existed the egg, which was an adaptation to help the survival of the species, for whatever reason.
If the question is taken literally to refer to a chicken, then as noted by others, then its the egg. As noted by MPL, the chickens ancestor is widely believed to be T-Rex. If thats the case, then the egg existed long before the chicken arrived on the scene.
http://www.usatoday.com/tech/science/20 ... -dna_N.htm