Search "egg packaging" in the search bar up top. You will get many answers to this question. Here's how I do it. Each egg is wrapped in bubble wrap, then placed into an egg carton. The carton is taped shut and then packed into a USPS priority mail box. I believe it is a 12"x12"x8", these are free from the post office. Around the carton inside the box I pack newspaper, packing peanuts, bubble wrap, or whatever I have on hand. The idea is to keep the egg carton stationary so that when the postal service kicks the box down the flight of stairs, regardless of the writing on it that says "FRAGILE, HATCHING EGGS, THIS SIDE UP, DO NOT X-RAY, etc", the eggs will be in their nicely packed and cushioned nest and hopefully make the ride unharmed. I've received many packages of eggs like this and the results were great.
I have also received eggs that were individually bubble wrapped, then those individual eggs were rolled "burrito style" into a blanket like tube of bubble wrap. These burritos were packed into a box as described above. I have had good results from receiving eggs packed this way also.
ETA: Fixed typos.
