Eliminating Illegal immigration will only be feasible if you also come down hard and swift on the businesses who hire illegals and the businesses who contract out to contractors who use illegal alien workers. A 10,000 dollar fine per illegal worker per day would be a good start. Make a cutoff date of six months from now and Actually ENFORCE the damn thing too.
Also stop handing out tax payer paid benefits to illegal immigrants as well.
It's not a business's (big or small) responsibility to police every job applicant, it's government's.