My take on it.
With regards to strictly chicken, chicken and egg came together. Egg is part of chicken. In order for the egg to become chicken, it has to be fertilized, which means you need a rooster. Where did the rooster come from with the egg? Without a mature rooster, egg is useless. Thus, chicken and egg must have evolved together.
Let's then move back a step. Birds evolved from dinosaurs. then the same question stays, which came first, dinosaur or egg? If we keep going back, we get to the stage of single cell organisms, which are the ancestors of all life forms on Earth. As I have mentioned before, egg is a giant single cell. So in a way (very loosely though), the ancient single cell organisms are eggs (very loosely defined). So egg came first.
However, with chicken and egg strictly, chicken and egg came together. egg without the chicken is useless and chicken without eggs is equally useless. they must evolve together.