Chickens evolved  from non-chickens  through small changes caused by the mixing of male and female DNA or by mutations to the DNA that produced the first actual Chicken Cell, or Zygote. These changes and mutations only have an effect at the point where a new cell is created. That is, two non-chickens mated and the DNA in their new zygote contained the mutation(s) that produced the first true chicken.

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 formed in the non-chicken's egg. So, the egg must have come first.