What about long term immigrants who end up disabled through no fault of their own? Or who get cancer? Or who end up having a child (who is an American citizen) and that child is special needs and the immigrant can't manage a full time job and care for their child? If they get pregnant and end up on bed rest or with a traumatic birth that takes them out of the workforce for a period of time?
There are ways to end up needing to rely on welfare that aren't due to laziness or a desire for handouts.
If the answer is 'kick them out', I'd be worried about what we're teaching our American kids watching. There are two lessons they could pick up, and neither is good for their moral development or sense of self. The first is that anyone who lacks the ability to work has no value, and that will engender greater alienation and isolation as they place all of their self-worth on their ability to earn money. They'll look upon the elderly, children, and caretakers with disdain (Interestingly, this probably won't help the birth rates either...). The second is that they are protected but those people should be disposed of when they're not useful. This will make them arrogant and introduce the idea of dehumanizing other groups, which will further the cracks of division in our society.
It take tremendous effort to immigrate, legally or illegally. Anyone telling you that they are lazy is obviously lying.