I'd suggest using a larger shipping box that isn't square, and putting some of those nice colorful "Fragile Hatching Eggs" stickers all over the box, especially on the top.
I just got two shipments of hatching eggs, one in a 8x12ish box with lots of those stickers on it. Inside were egg cartons, each egg spaced far apart, the top covered with plastic bags, then each carton was sealed tight with tape so the eggs wouldn't move. All around the egg cartons was balled up newspaper. I left a note for my carrier to leave the eggs inside my door on the floor, and she did. None of them were broken.
The next box to arrive was a 6x7ish square box, each egg individually bubble wrapped. The top of the box was too small for anything much to be written, so the sender wrote "eggs" in black marker, sort of small, on the sides, and the red "fragile" stamp was on the sides too. The same carrier, with the same note, didn't notice that this box was also eggs, and just tossed the box on it's side onto a chair on my porch. 3 out of 18 eggs were cracked in this one.
I feel like a little square box, not well marked with big pictures of chicks and very obvious labels, is begging to be tossed around. I'd expect a larger, flatter box to be handled better, especially with good labeling. If you want to build a good reputation in your egg selling, put a lot of care into the outside of the box, not just the inside, because a LOT depends on how that box is handled by the postal carriers.
Just my two cents!