There is plenty of demand for jobs in areas such as agriculture and the trades. If Americans want those jobs, they can get them. I worked in the trades for a while and I was pretty much constantly told that no one my age was getting into the trades, and its not because they don't have experience or training. I got in making a solid paycheck with no knowledge of the work whatsoever. Half my coworkers were living in a halfway home. No one else would hire them but the trades need the laborers. There is a pretty dire labor shortage even with illegal laborers.
A big part of the problem is that the US school system is pushing everybody into college instead of being focused on the demand for jobs. Illegal workers are typically filling a dire need in areas where US workers aren't picking up the slack, not forcing US workers out.
Further, while I agree that companies shouldn't be paying anyone below minimum wage (and many illegal immigrants do make above minimum wage), the purpose of the minimum wage is to protect workers. Getting rid of someone's income and sending them out of the country isn't protecting them. If the goal is actually to protect workers, it can easily be done through amnesty. Otherwise, I find the appeal to emotion to ring a bit hollow.
Yes the more Liberal leaning education system leans on College more than trades and aside from wasting money on College I am all for an education. The issue is that if you have the void being filled it goes away and has no need for change. If however everyone starts playing by the rules and a void in labor opens up it will be filled. I would leave my IT job in a minute for a good paying construction job, the problem is they are filled with low wage illegals. I know this for a fact because I have a couple buddies that are jobsite foremans and say that 90% of the people they work with are illegal. I was doing some welding for a company on the weekends and out of 50 employees not including managers, 4 were legal. Everyone else was illegal and not recieving benefits while taking a way lower wage than I was getting for the same work.