I don't know the entire process, but when I got shipped eggs, they were individually wrapped with bubble wrap, and then packed into an eggcarton, then packed into the box with lots of newspaper around it.
well there are threads on this, but here it goes, I do not do flat rate, but you can get free boxes from the PO, I use the 12x12x8 priority boxes, the rate I've had is between $8 to $12 for shipping depend where its going. then I place 1 1/2 to 2 inches of packing on the bottom i.e. plastic grocery bags, package peanuts, then place a sheet of bubble wrap on that, each egg is double wrapped in bubble wrap and placed in the middle of the sheet, all eggs in the same direction, keep them away for the walls, a second layer can be added the same way as this, when all eggs are in box pull up the sides of the bubble wrap and place one more sheet on top, place peanut or bags on the sides snugly, place packaging on top to fill up space, you want the box contents not to move be don't squish them, tape it up, address it and done
if you have any questions ask, plus there other ways too, cartons or not I prefer not to use cartons
I wrap my eggs either in thick foam or bubble wrap or alternate.
I use tissue paper, newspaper or some sort of paper to make it easier to remove so the egg slides out a-okay!
Then I wrap them in foam or bubble wrap or foam than bubble wrap.
Put them in an egg carton and tape it up and wrap bubble wrap around them and put them in a box with more bubble wrap and padding.
My boxes are often reinforced with cardboard and I use newspaper to reinforce the box sometimes too.
The eggs shoulud be snuggggggggggggged - I find that when the box is bulging, I have less broken eggs
in this case, less = 0. Usualy once in a blue I have a broken egg or two tops.