I'm not sure the H1B workers feel screwed, but they do tend to tolerate lower pay, more abusive working conditions, and poorer housing.
That's the problem. One solution would be strong unions that pay no attention to national origin, or if not that, at least more cohesive work groups with employees looking out for one another and refusing to be divided. Working people have got to quit attacking one another and turn their attention to the guys on top who are trying to divide them.
If someone is working in the U.S.A. then presumably they are contributing something to the U.S. economy and our society. Otherwise any talk of free markets and opportunity is a cruel farce.
I don't think where people come from is the problem. I don't feel any closer to an immigrant from Oklahoma than an immigrant from India. Their cultures are both foreign to me. My parents were both working in Hollywood when they met, my grandparents worked in the "high tech" California industries of their time, and before that my ancestors were dairy people and wild west ranchers. Their coworkers, hired hands, and occasionally their bosses were immigrants.
My last immigrant ancestor was a mail order bride to Salt Lake City. She didn't like sharing a husband so she ran off with a monogamous guy and they established a homestead that's still remote. Ultimately, all my ancestors came to the U.S.A. not for any golden opportunity but because their life in Europe was a hell of wars, religious oppression, and starvation. That kind of hell is a direct consequence of state religions and extremist nationalism.
At some point, if we continue down the path we are on, the U.S.A. will be so rotten that immigrants will stop coming and our best and brightest will start leaving. That's the equilibrium we've reached with Mexico. People with a decent jobs in Mexico don't want to move to the U.S.A.. The U.S.A.'s reputation as a "land of opportunity" is severely tarnished.
Turning the U.S.A. into an ignorant right-wing wasteland is one way to slow immigration, but it's not a desirable one.