I don't recall any one playing the angle of tax evasion - either charged to the employeer, illegal or both ! (You know, that's how the Feds got Al "Scarface" Capone!)
I don't imagine that many illegals have signed up with the selective service.
My first thought as well. Excellent angle.
I have suggested this as an angle in the past. Tax evasion is exactly what chaffes me about illegals. They put 32 dependents on their W4, use our social services and never pay any income taxes. In the meantime, they drive the wages down to the point an honest citizen who does pay taxes cannot afford to work at the wage offered. Instead, they sell meth to each other.
You and I are more likely to get crosswise with the IRS for trying to be a good standing citizen than an illegal is. The IRS has the data the government needs to fix the illegal problem. Instead, the IRS creates shadow files for SS#'s being used by more than one person and don't pursue invalid SS#'s. Nope, easier to audit you and I.