I think I read eggs should not be shipped via airlines, something about air cells rupturing with the pressure changes? I could be wrong but it would be something to research as shipping eggs would not be cheap and it can take weeks depending on the method of transport.
Live birds are probably possible if they were put on direct passenger flights as live cargo, most countries do allow animals to be shipped if they have health certificates, depending on where you are shipping from it could be VERY expensive. That is definitely not a situation where one wants to get the cheapest connecting flights.
Unless the buyer is extremely wealthy I don't see it as being realistic IMO. Also if the seller is in the US or EU I would think it could introduce new diseases (i.e. latent Mareks, etc...)