I cannot speak for other states, I only really know the rules here in Oregon.
It is not cheaper for an illegal mexican to build a fence for the government. We have prevailing wage here. I could hire you for 14 dollars an hour to build my person fence or the fence for my personal business but if you are going to fence anything that has to do with government, from schools to courthouse to anything else government, you are going to get prevailing wage. (three times or higher of the normal wage)
In my experience in Oregon and this dates back a long ways on this topic, illegals are not working for big companies, they are mostly working for small businesses.
When I was a kid we helped illegals come over, helped establish them by getting them housing and work. But back in those days they were very mistreated by mainstream Americans, taken advantage of and they only came to tree plant for 6 months then go home with the money they earned to take care of their families. They almost never brought their families but instead sent the money home. They were literally doing the job Americans did not want to do. This is not the case anymore. I do not offer help neither does my family anymore.
Most of the illegals I see working here are doing dry wall exc.
or small companies. and some have gotten loans from Wells Fargo to start their own landscaping business, roofing business, window washing business exc. These are jobs Americans would do. What big companies do you know that hire mostly illegals? It could be the case in other states but I sure do not see it here in Oregon. I know in some mid west state Wal-Mart got busted for hiring illegal alien janitors but it turned out that Wal-Mart contracted a legal business to clean and that small business hired the illegals, not the big company Wal-Mart.