Prior to the first true chicken, there were non-chickens. The DNA changes came about in cells housed in the egg. So the egg came first. In July 2010, British scientists, using a supercomputer, claimed to have come up with the final and definitive answer.
So in a nutshell (or an eggshell, if you like), two birds that weren't really chickens created a chicken egg, and hence, we have an answer: The egg came first, and then it hatched a chicken.
