Many, many people are having successfully hatched eggs that were shipped in bubble wrap. Obviously it doesn't suffocate the eggs. If it did, how did all these eggs hatch?
The only successful hatch I've had with shipped eggs, they were heavily wrapped in bubble wrap.
However you wrap the eggs, if they have room to be jostled at all, (not just "too much", any movement is "too much") they may break, or have severe air cell damage. They need to be immobile. The padded materials need to be good at absorbing shock. There will be impact shock applied to those packages, so eggs must be packed well enoughto protect them from that. This gives the best chance of arriving intact, with good air cells. A unbroken egg with a broken air cell has a slim chance of hatching. An unbroken egg that's scrambled inside, has no chance of hatching.
As for flat rate boxes, they're not deep enough to protect the eggs well enough IMO, and the postage for regular priority boxes was actually cheaper than flat rate, when I shipped eggs recently. Except once, when I shipped enough eggs that they weighed enough that flat rate would have been cheaper. But with that many eggs, flat rate would have been packed way too full. So far all my eggs arrived at their destination, intact.