I always took the question to mean "which came first, the egg-birthing process, or the complex animal that reproduces via eggs"
Because, in all reality, the "egg" outdates "chickens" by (possibly) hundreds of thousands of years - the chicken wasn't the first animal to lay an egg!
... but if we change the question so that "chicken" means a significant specie... then... an egg must come from a pre-living organism...
I was always curious if it's possible to argue that egg-like means of birth (i.e. leaving your young out in the wild with a parent-provided source of food for their development) was the first major way of creating descendants?
Can we consider the first time a cell (of any complexity) replicated itself as an egg? or is that a chicken?