OK, here's what I would do: Bring the eggs with you to the airport. Be as low-key as possible. When it's your turn, hand the egg container to the TSA offier NEAR the X-ray and tell them with a smile they're fertile and not to X-ray them but that you are taking them through the metal detector - if they look like they don't understand what to do, just ask "hey, what do you do with small dogs and cats that travel in the cabin?" Comply with every "pat down" or whatever they throw at you WITH A SMILE, DO NOT LOSE YOUR COOL, stay friendly and try not to seem weird.
Bring along a printout of some email correspondence with the egg breeder, or something on paper that backs up your story (printout of the place you got them, etc).
If the TSA is getting bent out of shape, just calmly think "collaboration!" and take the POV of "let's try to solve the problem TOGETHER," tell them that when you called ahead they instructed you to do exactly this (even if that's not really the case, it will make you seem responsible and the local TSA won't be able to fight you on it), and if all else fails ask for the TSA supervisor NICELY to request her ideas for how we can all handle this so you a) get on the plane on time, b) your chicks survive without getting cooked in the X-ray.
As long as you remain calm, keep "trying to help them help you," and seem friendly and above all, NORMAL, you'll get on the plane with your chicks. Remember, they are looking for the bad guys; any clues to which you are mentally unstable and/or a religious extremist - not a chicken enthusiast.