No minimum wage + No illegal immigrants = Republican Dream Scenario???

It seems to me that the immigration issue and the minimum wage issue go hand in hand. If you can eliminate the minimum wage and eliminate illegal immigration, you can have a nation of tax paying citizens that are being payed above board at the rates that illegal immigrants are currently being payed. In this new scenario companies would now be able to take advantage of low cost legal employees and whatever tax exemptions or benefits that come with it. Do republicans really want to reduce or eliminate the minimum wage to force Americans to work for less pay so republicans can line their own pockets and bank accounts? Is this what they mean by "free market"? What effect will this have on the country? I've read a lot about raising the minimum wage and its effects on immigration, but I haven't seen anything discussing the minimum wage immigration issue from this point.