For starters, there are illegal aliens who get a social security number. And once they get a social security number, they have the social security payments withheld from their paychecks. If they are only paid in cash and there is no paper trail well that's a totally different story. So misconception #1 illegal immigrants have never contributed to social security is false.
Misconception #2 really needs to be addressed. People believe that they have paid into social security similar to the way you pay into a 401K or IRA. THAT IS FALSE! Social security has NEVER worked like that. Social Security is really more like a Ponzi Schemeusing the the "rob-Peter-to-pay-Paul" principle to work. The current labor force (who are paying social security taxes) is supporting all those who receive their check. (Side though: I wonder what happens when the labor force is smaller than the recipients. Hmmm. Oh wait that's a few years away). So when you get the mailing from the social security administration that looks like an investment statement explaining how much you have paid, realize this is misleading and a waste of your tax dollars.
So to review: Just because a worker is an illegal immigrant doesn't mean they haven't paid taxes including social security. And you are not paying into anything when it comes to social security. The statement that it is impossible to live on social security, however, is true.