Illegal aliens are not eligible for most benefits, but they are eligible for some. I had a client, whose entire family was here illegally; however, because she was a disabled child, she received (expensive) medical coverage. A woman who is here illegally and gives birth to a child will have her hospital bill paid by Medicaid; the hospital even submits the application for her.
Furthermore, a family composed of 2 illegal parents and children born in the U.S. can apply for food stamps for just the kids. They usually get the maximum benefit because they have no real way of verifying their income...they just turn in statements from their employers who pay them under the table and can write down anything about what they actually make.