You can mail eggs, hatching eggs are mailed by the thousands. You probably want to send them at least Priority and be sure you package them well, the USPS is not happy if the eggs get crushed and the package leaks.
Off the USPS website.
5 Perishable Matter
51 Definition
Perishable matter is anything that can deteriorate in the mail and thereby lose value, create a health hazard, or cause an obnoxious odor, nuisance, or disturbance, under ordinary mailing conditions. Mailable perishable matter may be sent at the mailer’s own risk when it is packaged as required and when it can be delivered within appropriate and reasonable time limits to prevent deterioration.
Examples of perishable matter include mailable types of live animals, food items, and plants.
54 Eggs
Eggs are mailable in domestic mail subject to the following general conditions:
They are individually cushioned.
They are otherwise packed to withstand shocks encountered during normal Postal Service handling.
They are not likely to be harmed by anticipated temperature changes while in Postal Service custody.