Why hello. Did you call for me? It's like when you say Beetlejuice three times.
This is what I like to call creative attempts at collecting if you fail to pay your bill. No, you don't have to enter your credit card information. Just like you don't have to enter your SSN. You also don't have to sign the release of all of your medical records. (I don't sign theses for a number of reasons, plus they don't want them. It would take a uhaul. I prefer to get what they want. I am very tight with the front office at my doctors' offices and I am happy to pay the small fee for copies and their time.)
The main reason for obtaining your social is for collections purposes. If they have to sue you for money owed, it is a whole lot easier to get a garnishment if you know a person's social security number. (I have been on the suing end of things for many kinds of businesses.)
You don't have to fill out anything that you don't want to. Plain and simple. On any form. Your social security number is considered protected personal information during legal proceedings. I would also argue that a credit number is protected private financial information.
For fun, if the front office tries to pressure you, I'd have a bit of fun and say, well, okay, but I am uncomfortable providing this information, so I'd like my lawyer to review your forms first, so can we reschedule for a few days from now. I suspect that any pressure will be quickly alleviated. (This is especially fun for contracts related to automotive services. Tell me when they would need my SSN in any context???)
However, if you are unwilling to provide the information, the service provider may reserve the right to refuse service. Unless the person is the top in his or her respective field, you can probably find another pretty easily. In Vegas, there are dentists in every shopping center. Some will sedate you before the procedure. Some will lie and say you have six cavities even though you have never had a cavity (happened to me). Some will lie and say you need a root canal when something is just jammed way up high in your tooth (also happened to me). I personally don't want any of them having my credit card information. I can't even trust them to do their jobs.