We'd need to find exactly where it "passes over", which could depend on who you ask.

No, we don't. It doesn't matter when that is, because you and I both agree that it's out there somewhere, and that at the point in time referenced, a non-chicken laid an egg and a chicken hatched out of it. That's all we need out of that point, and neither of us are disputing that part of it.

If you define a chicken as hatching from a chicken egg ("every chicken must have hatched from a chicken egg"), then the egg came first. If you define a chicken egg as an egg that was laid by a chicken ("all chicken eggs must have been laid by chickens"), then the chicken came first.

Agreed. I, personally, use the broader egg definition you reference in the last paragraph, but a definition of "chicken egg" would put the whole thing to rest, and I propose this: Not every chicken egg contains a viable chicken. We all agree that these eggs are still chicken eggs when we buy them at the supermarket, though, so my proposed definition is that a chicken egg is laid by a chicken. Otherwise, we end up with unclassified eggs in our omelettes, and we can't have that.