Start politely, explaining your position and expecting them to understand and cooperate. Talk to people higher up in the organization. This often works simply because the organization has a standard way of dealing with requests not to use the SSN, and the first person you deal with just hasn't been around long enough to know what it is. You have to decide whether talking to someone in personnel, and possibly trying to change corporate policy is going to get back to your supervisor and affect your job. The people in the personnel and benefits departments often carry a lot of weight when dealing with health insurance companies.
Most newspapers can get a quick response.
If it's a federal agency, your congress member may be able to help. If you're talking to someone who has some independence, and they'd like to help, they will sometimes admit that they know the reason the company wants it, and you can satisfy that requirement a different way. If it's a case where you've gotten service already, but someone insists that you have to provide your number in order to have a continuing relationship, you can choose to ignore the request in hopes that they'll forget or find another solution before you get tired of the interruption. If anyone else used your number when applying for a job, their earnings will appear under your name in the SSA's files. As a result of a recent court case, the SSA has agreed to accept corrections of errors when there isn't any contradictory evidence, SSA has records for the year before or after the error, and the claimed earnings are consistent with earlier and later wages.