First of all, becasue they are illegal thos mexicans are not hired for jobs that would keep our industrial capacity here. Secondly, you are right about the risiding on the taxpayers bill, which is why we need to stop giving welfare to these people. The ones we want come here to work, not to collect a check.
The Chinese workers power to negotiate exists in their freedom to not take these jobs. Please read Friedman, you will understand what's going on around you far better. The "It's not fair" school of economic thought never got anybody anywhere.
When you're starving in the country side, living on dirt floors, that's not much of a choice and when you're in a prison then you really don't get a choice. Even the military is in on it. Soldiers spend half a year working for free at the generals' factories.