@Just-Lisa-Now- I've never seen someone with a green card but a Social Security card showing not eligible to work.  I think that's barking up the wrong tree, although maybe a new card was not issued once the green card was issued.  I'm not even sure where I would input "not eligible for employment," but then I don't see many new clients and I generally don't ask to see their Social Security cards.  

The USCIS website states, "Having a Green Card (officially known as a Permanent Resident Card allows you to live and work permanently in the United States."  (No, they did not close the parentheses.)