The iron law of wages says that wages will be pressed to the floor. That is what is happening to the glee of the US Chamber of Commerce.
I have no idea why you are talking about the "iron law of wages" in relation to 10 million people leaving the workforce. The iron law of wages basically says that an employer will pay a worker the bare minimum in order to keep them employed. The iron law would apply to the illegal immigrants, not to mainstream Americans.
I might as well lay out my reasoning for why I think wages would stagnate and/or go down if 10M people were to leave the economy.
The exodus of this many consumers would mean there would be an incredible excess amount of goods available for purchase. Higher supply, lowered demand. Prices on goods would fall. This is deflation.
Not only would prices fall but people would put off buying stuff expecting prices to continue to fall. They've already dropped 20%, lets say, why not wait until they drop to 30%? People would slow their spending.
This is the trap of deflation. In this sort of an economic condition, where people are holding off buying things one of the last things that would happen is wage increases. If products aren't selling, meaning companies aren't making good profit margins and earnings are depressed, why in the world would they increase wages? Where would they get the money to do that if their products aren't selling and earnings are down? It's also likely that people with manufacturing jobs and other sectors are having reduced work hours. Wages are not going to go up in this environment.
Now, there would certainly be a shortage of minimum wage workers. Yes, business owners in these sectors would have to increase wages to lure people to take these jobs. But these are crappy jobs. No one is going to leave their job as a dental assistant to flip burgers (for example). Increasing wages in these entry level jobs would increase prices on the products produced by these jobs. If most things in the economy were getting cheaper but certain fruits and vegetables were getting more expensive, or the price of re-roofing your house was going up, or fast food was going up, or the cost of a hotel room was increasing, for example, consumers would simply not buy as much fruit, wouldn't get their roof redone, wouldn't go to fast food joints, and would curtail traveling.