Sure it is. Fines and jail time for those who knowingly hire invaders, or don't do thorough checks, would close the job market. No jobs, no free medical, food stamps, housing etc., the invasion would be curtailed.
Besides that, these aren't "jobs US citizens wouldn't do". They would do them, if wages, hours, and benefits weren't at the slave level.
Interesting point. Do the freebies attract people? A few, maybe, but overall it wouldn't make a bit of difference. Where they come from all their property is extralegal, all the work they do is extralegal, all the businesses they own are extralegal. They won't do any worse inside the US even without the welfare--that's just bonus.