Deport them back the POS country they came from. If they want to come back LEGALLY then they are more than welcome at my church. But they better learn english because we don't do bi-lingualism.
Just as many have written, generally speaking, there is no obstacle to solving all this.
Go after the employers. If illegals are unable to work, they will self-deport. Before the welfare state emerged, this happened each time the U.S. went into an economic plunge.
To minimize the welfare state's safety net, require applications for assistance to be processed through INS. Tell hospitals and clinics, yes, they are responsible for proof of citizenship. Tell them it is a form of health insurance.