Exactly. So there's no way to measure the exact egg that was first born to a species we would not recognize as a chicken.

(Edit: Warning: Only bullshit meant to amuse and fascinate follows. I've been watching too much "SmartyPants" on DropOut.tv, where they try to make each-other laugh with serious sounding silly presenations.)

Further, we might each choose a different arbitrary egg and declare that eggs parent "not a chicken".

But for this question, that doesn't have to matter.

If we can all agree that something in the ancestry of the modern chicken was not a chicken, and agree that it was likely still birthed from an egg, then we can conclude that that egg came first.

Even if we cannot agree about which exact egg hatched into the first chicken, or which exact animal was the first chicken, we can agree on their relationship such that we can agree that any selected "first chicken egg" came before any selected "first chicken" to be born from it.

The hardest part of this proposition is whether we can agree that the first chicken was born inside an egg. I propose that it must have been, by our own definitioms, because we widely agree that chickens are born from eggs. Not by any intrinsic property, but simply by our accepted definition of the word "chicken".

So any hypothetical chicken-ancestor we choose as the "first chicken", but not born from an egg, we should not be willing to call "first chicken", after all.

So we must proceed forward in time from that failed choice of "first chicken" until something sufficiently chicken-like is born from an egg. Then we can call that animal our "first chicken", and examine it's relationship to "chicken eggs". We will, by our method of searching, always then find that the "chicken egg" that our "first chicken" hatched from, came first.