While the US Citizenship and Immigration Services (USCIS) remains operating during the government shutdown, E-Verify services are not available.  This means that companies cannot enroll in E-Verify, and those already enrolled cannot verify employment eligibility, make changes to an account, or run reports.  The USCIS has provided limited guidance on their site, which you can read here

Employers enrolled in E-Verify must continue to complete Form I-9 for new hires during the shutdown, even though the three-day rule for verifying employment eligibility online is suspended.  When the shutdown is over, employers will be able to verify the employment eligibility of those hired during the shutdown.