Problem 1: Illegal immigrants.
Problem 2: Americans who can't find a job.
This seems like a matching problem. I propose that DHS crack down hard on employers of illegal immigrants. To solve problem 2, they must coordinate with local unemployment agencies. DHS will arrest and deport illegal immigrants. For each illegal thus deported, the unemployment benefits of an American (maybe whoever has been on unemployment the longest) will be terminated and they will be ordered to report to work at the former employer of the illegal. Employing a formerly unemployed American will be part of the penalty for the employer of the illegal immigrant.
To induce cooperation among employers, we could create an amnesty program. Employers can report their illegal workers to DHS and avoid paying fines. Maybe DHS/Unemployment Office will even give extra help to minimize disruption to your business (i.e., no deportation until the day replacements are available).
[edit: to those downmodding, just curious. Is there an obvious flaw in my plan that I'm missing?]