If a client doesn't own a domain name and currently I don't offer registration can cPanel be set-up to user.hosting.com instead of www.mydomain.com?

I think that this can only be done if they have control of the www.mydomain.com CPanel account. But if you made some DNS changes I'm sure you could get it to work.

Say cPanel is setup on domain.com. You can make a new account under name.domain.com. HOWEVER, when transferring files with cPanel, you'll be in a whole mess of trouble, since client files will be stored as a subdomain during the move, rather than a regular domain. So just choke up the ten bucks and get a regular domain name -- save yourself the trouble.