Read about the EEOC litigation levied against users of E-Verify. The two seem at odds such as to place an employer in a difficult position in applying E-Verify to reject employment. Raising questions about legitimacy of documentation often became such a no-win for the employer, that fake documents became no longer challenged.
This terms within this policy need serious scrutiny. Conflating illegal aliens and immigrants has allowed an opening for foreign nationals to manipulate the court system and gain leverage per hiring practices, against the employer.
https://www.eeoc.gov/laws/practices/inquiries_citizenship.cfm