Comment by spiralpolitik

If you are a non citizen, you should have "Not valid for work without authorization" written on your social security card. The social security administration knows who is and isn't a citizen because it's the part of the government that's reviewed the paperwork you have proving your status (birth certificate, naturalization document, work authorization etc).

(When you become of US citizen one of the first tasks is to take your naturalization document to your local SSA office so that they can update your file and issue you a new card without the notice)

The only citizens who aren't in the social security database are people who were never registered at birth, either because their parents didn't tick the box when registering or they were born overseas and their parents never registered the birth with the US government. Both can easily get registered via their birth certificate.

Everyone else in the system has already provided documentation to verify their status one way or the other.