Sorry, but you are wrong.
Illegals ARE being hired - especially on the really big construction jobs that get municipal, state, and federal dollars/tax subsidies. In fact, the Latinos (illegal and legal) outnumber other ethnicities. And, they are getting union/prevailing wages.
While they may be incidents where illegals are being hired, you can not back up the claim that illegals are being hired universally with full pay and benefits.
And if they are "on the really big construction jobs that get municipal, state, and federal dollars/tax subsidies", they in violation of a lot of laws besides immigration that I doubt many firms would want to risk violating.
If you can prove this, send me the names of these companies and I will report them. I don't think you can or will.