You might be looking for a long time if your company drops you because you refuse to supply the information that they request. Actually it might not matter, because all the states are now requiring your social security number when you renew your license. I believe that it is part of the Homeland Security Act. When your insurance company checks your driving record, they might be able to obtain it from the state records. State Motor Vehicle Departments often share information with insurance companies that they wouldn't share with the public because of the interaction between the state and the insurance companies. In many states, if you cancel or allow your insurance to lapse and don't turn in the registration plates the insurance company sends notification to the state. In CT, if you don't turn the plates in before the insurance lapses, the state will hit you with a $200 fine. It is one of those things that you either pay or you don't get to register a car until you do.
Another thing that MA & CT both do, if you are a passenger in a car that is stopped for a motor vehicle violation, they might run the drivers license number of all passengers and if they don't have a drivers license, then they ask for your social security number and run that. I don't know what happens when people refuse to give the information. Usually it is the younger people that don't look "family oriented" that have the problem. After 9/11 a lot of things have changed and many people are not aware of the changes until they or someone they know has had an encounter with a police officer that is following every directive to the nth degree.