If someone is using a SSN as proof of citizenship, they are blatantly wrong. SSNs can and are issued to aliens.
https://www.ssa.gov/pubs/EN-05-10096.pdf
USCIS lists as common proofs of citizenship: US birth certificate, US passport, certificate of citizenship, or naturalization certificate.
https://www.uscis.gov/sites/default/files/USCIS/Resources/A4...