You are right that illegal aliens cannot collect SS. But they do collect SSI and SSDI.
No, they cant. Illegal aliens do not have a social security number which you need to file for any of those programs. I know this because my buddy works for SS.
Now what does happen is these illegals aliens hook up with some American woman and have kids. One of those kids is disabled. They then file for Supplemental Security Income which is a needs based or welfare disability program for those Americans that did not pay enough into the system to be eligible for Social Security Disabilty and for children whose parents fall under certain income levels.
From what my buddy tells me its a huge pain in the ass when one of these cases comes in. He had a case where the child had cerebral palsy and the dad was an illegal. The SS computer system has to have a social security number for the parents to record their income. He says they have to do some sort of work around when this happens and it is a headache and takes a lot of time.
In case your wondering, no they are not allowed to call ICE in those cases. Pisses my buddy off to no end.