I believe that E-verify should be used by all employers. I believe employers should be doing and excessive background check to see who they are hiring. People who commit crimes should not be ahead of anyone who has lead a clean life. Also, we should know who has done what especially in jobs involving children.
Though it is true the E-verify is able to flag undocumented immigrants, About a million people have been flagged as illegal even though they were legitimate us citizens. It might be able to prevent undocumented immigrants from getting jobs but that will lower the economy and also prevent US citizens from getting jobs.
Finding a job is hard enough as it is. Potential employers already have access to our Facebook and Twitter. They can pay for credit and background checks. That is enough information right there. They can find out as much information as they wish too already. They should just be prosecuted if they are caught hiring someone they legally can't hire.