If they are guaranteed a higher wage, who is going to break the law to hire them? There is no reason to do so at that point. The only thing that makes these people appealing is not having to pay them fair wages.

You think businesses, saddled with higher minimum wages, will be less likely to seek out employees whom they can abuse in other ways by leveraging illegal status?

It wont be worth it for them to hire illegal immigrants at that point, no matter what. They will become too expensive.