If I had a UPS store in my area, I would so do that!! You should put some wood shavings, pillow stuffing, napkins/paper towels, or something inside that carton to keep the eggs still so they don't bump each other or bounce around inside the carton. Most breaks are either from hitting each other, or being too close to the side of the box, and being hit from the outside. The UPS store has all the boxes and fillers you would need there, of course they would charge you to pack it.
Otherwise, you can do it the way that adkins90 said. I believe that the prices to ship would be about the same for UPS or USPS. You can also go online to usps.gov and order boxes to be shipped right to your house. The boxes and shipping of the boxes are free, if you use them for shipping through USPS. There is also tape and priority labels that you can order.
I have had most of my eggs shipped to me by USPS, and one by UPS. None of the UPS eggs are developing, they are all sloshy. I have ducks, quail and chicks from the USPS eggs.
I just have the buyer send a money order, since I always have eggs, they never get old. Whatever is not used for hatching are used for eating, so it is no big loss. When I finally get my bantam flock underway, I'll have to figure out how to accept payment through paypal.
Hope this helps!!
Also, if you use USPS, it is usually cheaper to ship priority non-flat-rate, depending on where you are shipping to.
ETA: Sorry, it's usps.com, not .gov silly me!!