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. Enlist the aid of your employer. 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. Threaten to complain to a consumer affairs bureau.
Most newspapers can get a quick response.
If it's a federal agency, your congressmember may be able to help. Insist that they document a corporate policy requiring the number. When someone can't find a written policy or doesn't want to push hard enough to get it, they'll often realize that they don't know what the policy is, and they've just been following tradition. Ask what they need it for and suggest alternatives. 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.