The egg came first. This is because we have all evolved from dinos so has the chicken. so it started as a dino then started evolving into a chicken then finally one day a dino/chicken egg hatched and out popped a chicken ......... POP!!!!!!!
A chicken as we know today would have at one point been a mutation from another close species. Just because a chicken hatched from the egg does not mean that the egg was laid by the same exact creature (just something VERY close, yet different). The egg came first.
According to the Creator of chickens, and the author of the Record of their origins, chickens came first. It was on the Fifth Day of Creation Week that He created "every winged fowl after [their] kind" (Genesis 1:21) complete with the DNA to reproduce that kind. Then He "blessed them, saying, Be fruitful, and multiply" (v.22) using that DNA. For the chickens this meant lay chicken eggs. Problem solved.