Out of status, yes. Ineligible for employment, yes. "Illegal" is slang. Either you entered without inspection,are in status or out of status. But the end result for you is the same. You are removable if you do not have legal status.
NO! Social Security Number is issued by the Social Security Administration (SSA) and not by Immigration. Having one make no differences for immigration purposes.
Someone who is an illegal immigrant will be considered such even if he or she managed to get a social security number.
NYC EXPERIENCED IMMIGRATION ATTORNEYS www.myattorneyusa.com; email: email@example.com; Phone: (866) 456-8654; Fax: 212-964-0440; Cell: 212-202-0325. The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter.