I've never shipped birds, but have had plenty of them shipped to me. I've gotten several boxes of day old chicks from hatcheries, and they all ended up healthy. Also had six 1 month old silkies shipped to me from Florida in November, and one 3 year old silkie hen shipped to me in January. All of them made it and are healthy. My best advice for receiving poultry in the mail is to give them Nutri Drench afterwards, and make sure they're getting plenty of water.
To ship chickens, you generally need NPIP certification. Getting it is a different process in each state. You'll need a poultry shipping box, with air holes and lots of 'fragile' markings. Fresh fruit is good to put with them, as it keeps them hydrated and full. You should drop them off at the post office in the evening, and then pay for overnight shipping so that the customer gets them early in the morning. This way, they don't go without water for much longer than they would any other night. Have them hold it at the post office, and then call the customer to come pick the chickens up.
Good luck!