I think dipsy doodle < goodness now i cant remmber the name for shure> anyways I think they have pictures on there byc page on how they shipp eggs .. Good luck as far as I can tell it involves lots of bubble wrap and packing filler .
You can get the boxes from the post office. I wouldn't recommend using the flat rate unless you are shipping a large amount. The post office doesn't have a very large selection, but if you go online to the usps website, you can order all sorts of boxes for free.
DipseyDoodleDoo has a link under her name on her BYC page of how she does it. Just remember, lots and lots of bubble wrap!