In nature, living things evolve through changes in their DNA. In an animal like a chicken, DNA from a male sperm cell and a female ovum meet and combine to form a zygote -- the first cell of a new baby chicken. This first cell divides innumerable times to form all of the cells of the complete animal. In any animal, every cell contains exactly the same DNA, and that DNA comes from the zygote.
Prior to that first true chicken zygote, there were only non-chickens. The zygote cell is the only place where DNA mutations could produce a new animal, and the zygote cell is housed in the chicken's egg. So, the egg must have come first.