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Depends on how you want to ship.  You can use flat rate boxes from the post office, which are free, or the PO sells "ready post" boxes.  You can get bubble wrap either from the PO or from a store like Kroger, or maybe Target?? Or a UPS store would have everything, though I am sure it would cost more there.  Maybe talk to a vet clinic about saving packing materials for you?  That's a good source because they will get lots of small sturdy boxes, and all of our glass bottles come wrapped in bubble wrap.
Personally, I am really lucky because I work in a veterinary hospital, and the girls that unpack our boxes of supplies save all the bubble wrap for me, and we already save all the smaller boxes, so I just pick through them as I need them.  But if you go to the PO and see the thickness of the flat rate boxes, that will give you a good idea on the sturdiness of the boxes you will need to be on the lookout for.  
When I ship eggs, I make a little "nest" in the box out of wadded up newspaper, with some shredded paper (or something similar, depends on what I have handy) in the bottom.  I then put down a piece of bubble wrap (I use the kind with big bubbles for this part) into the box.  Each individual egg gets bubble wrapped (regular size bubbles) with about three layers, and nestled snugly into the "nest" (don't want them banging around). Then the ends of the big bubble wrap get folded over the top of the egg nest, and if there is extra room in the box, I fill it with shredded paper, or whatever I have used in the bottom of the box.  Making sure everything is snug and nothing jostles around in the box. 
Hope this helps. 
Oh, and with this packing method, out of all the shipments I have sent out (even to Alaska) I have only had reports back of 2 broken eggs ever.