The second criticism is that it will "force" illegal immigrants into the underground economy. This is probably true. However, jobs in the underground economy are significantly less attractive to potential immigrants. Therefore e-Verify can have a substantial effect in reducing future illegal immigration.
Unlike Italy, the U.S. is willing to prosecute serious operators in the underground economy. Not the homeowner who hires a street corner guy to help with occasional yard work, but the garment factory or slaughterhouse with many off-the-books workers.
The article says self-deportation won't happen, which shows a serious misunderstanding of our illegal immigrant population. Most illegal immigrants are from Mexico and many of these go home to Mexico every year to visit family, start new families, etc.
In a situation where work in the U.S. becomes unattractive and poorly remunerated, the motivation to return to the U.S. is seriously reduced. Self-deportation can work, the question is to what extent. That remains to be demonstrated.