I work at a dental office. Most insurance companies still require a Social Security number in order to process claims. If the person refuses to supply a SS#, we tell them to file their own claims. When we tell them that, they usually supply the SS#.
Nobody has a right to use your social security number for any reason other than income tax. Of course, it can make life difficult for all the industries that continue to require it (e.g. credit agencies, insurance, etc.). Heck, I remember when Holiday Inn used to use SS# as their Priority Club ID. Nuts!