I believe that hatching eggs are customarily shipped via Priority Mail. I recommend doing an eBay search for hatching eggs to learn more about how this is done. Most sellers provide a description of how they collect, store, package and ship their eggs, so you can learn a bit just by reading those listings.
Eggs can be shipped express, priority, or parcel post. The reason why most people use priority mail is because it normally takes 2-3 days for delivery and the price is reasonable. Express now takes up to 3 days it seems, and the price is outrageous! I personally would not pay the extra shipping to have eggs shipped express.
I agree with Shelly. Express on my chicks & birds has been abysmal lately. Everyone has made it alive, but that's because I provide a TON of wet feed & fruit. I could have saved a fortune and sent them Priority. Priority Mail, in most cases, is fine for hatching eggs.
Fedex offers overnight, 2 day and express saver. The express saver is 3 day and is USUALLY close to the price of a flat rate box....UNLESS you live somewhere in the boonies. The only problem with flat rate in the service for the post office has been seriously slipping lately.
I shipped eggs from Colorado to Maryland Fedex GROUND and it got there in 3 days with only 1 cracked egg out of 14.