Hi! I'm pretty sure you can go in and just ask for a new backing...It seems as if the SA's usually have a mini stash of backing kept on the chore just in case. I'm not 100% sure but I would definitely try! Sorry if this doesn't quite answer your question!
I'm not sure since I have never lost a back (as of yet). I would go to wherever you purchased the earrings (boutique, NM, Saks, etc) and ask about getting a replacement back.