Yes, a SSN is not necessary to place an eviction judgment on someone's credit.
You may have some remedies available to you if the unit is illegal. Contact the local health or building department, or a landlord tenant attorney.
See the link below for info on your rights as a tenant.
Yes, they can still evict you. However, as Mr. Spiritos mentioned, since your unit appears to be illegal, you need to contact the local housing department and find out what your rights are.Ask a similar question