Businessmen don't just want cheap labor.
They prefer undocumented illegal people who won't join unions, sue their bosses, agitate for better working conditions, or go to another company to negotiate a better deal. And they have the added advantage that you can always turn them over to the Migras on payday if you don't feel like paying them. Republican immigration law isn't designed to keep illegals out, its designed to keep them as illegal as possible.
These people are desperate, they'll do whatever it takes to feed their families and stay alive. But they are not stupid. They know who is screwing them, and why, and one day they WILL get even. And those of us who never hired an undocumented worker will suffer too.