Who private companies hire shouldn't be up to the government. That's not fair at all. I mean, there are restrictions, such as child labor laws and such, but aside from that, the government absolutely should not be involved in the affairs of a private company. That's way too much power open for loopholes and abuse.
That being said, it is 100% illegal for anyone, immigrant or citizen, to be paid under minimum wage. Companies simply don't do it because when and if the government finds out, the company pays a fine of all the unpaid labor and then some. It's simply not financially viable to lose a worker on top of the amount of shit they'll be put through by the government -- constant inspection, losing a worker, the possibility of budget cuts due to lost funds, etc. It just doesn't happen as much as you think unless it's an unofficial, payment-under-the-table company that pays in cash for extremely short hours of labor, which other Americans wouldn't take anyway, because those companies don't follow labor laws. It would be the same if a family hired a kid to mow their lawn for $5 a day.
Oh and did you know immigration won't come out and investigate a place with illegal workers unless there are several complaints?
That is so not true it hurts. It may be something that personally happens in your region, but here in South Florida, they're absolutely up your ass about hiring illegal immigrants. If someone even speaks in a tone of voice that could HINT at a complaint, you have immigration people showing up at your comapny's door.