Did not provide SSN, will charge show on my credit report?

I rented a book from this company online and upon returning they claim the book was damaged. I am 100% I did not damage the book so I canceled the credit card they have on file so that they don't charge me a blown-up price for a replacement (plus who knows what other made up charges). I know in a little while they will be start sending me in bills in my mail. If I choose to ignore them can they eventually show up on my credit report? I never provided my SSN to them.