There are many different methods to shipping eggs; the best success I have ever had was an egg carton, with bubble wrapped eggs (each egg) and bubble wrapped egg carton with packing peanuts all around, shipped in a priority mail flat rate box.
There are also foamy thingys that are available; I think bubble wrap is cheaper.
I can tell you how I received eggs from bargain. Each egg was wrapped in bubble wrap, and packed into a box snugly with popcorn, bubble wrap and misc styrofoam. And there was a foamy padding on the bottom of the box. She then put that box in a larger box and padded it.
The eggs made it all unbroken from GA to CA. They are in the incubator right now. (Day 6)
There are a lot of threads already on here about shipping.. you might try the search engine.
I've used cartons, and not had very good luck. I used to bubblewrap each egg, then nestle it snugly in crumpled newspaper. But then I received eggs from OnTheSpot, and she just used newspaper. It was the best ever, no breakage. I use the method now (wrap each egg like grandma's fine china then nestle snugly in crumpled newspaper) and haven't had any broken eggs, even the ones from here to Florida.