SSN is not a proof of lawful employment authorization. Under the U.S. law, SSN is used for taxation and other legitimate purposes other than confirming lawful employment authorization.
(The U.S. Department of Homeland Security's E-verify system use employees' SSN to verify the status with Social Security Administration. However, if the record does not show, it does not give the right to employers to terminate the employees with issued, as employers are required to allow employee to correct the problems.)
If you don't have the document listed in the following form: