Maybe I'm reading it differently... If you have to have a SSN and U.S. Citizenship or legal presence how can an illegal get one?
If an immigrant has a green card, he can get both a SSN and a drivers license, without US citizenship.
Not long ago, a foreign student could get a SSN. The SS card just says "Not valid for work.". That means he could get a drivers license in Virginia. And if his student visa expires and he decides to stay in the US illegally, now he's an illegal immigrant with a Virginia drivers license.
No, a drivers license is no proof of US citizenship or legal US residence.