Can an employer file for labor cert after non confirmation on e-verify?

They can, but if awarded, the alien's illegal status is highly relevant to whether the petition and visa/adjustment are approved. If you accumulated any illegal presence, then you may be denied a visa and/or application for adjustment, and/or arrested.

Thanks for the response. Please can you explain illegal presence, I thought that it meant if you left the US then you are barred for a number of years depending on the number of years you overstayed the visa. Thanks in advance.

If you overstayed 6 months to one year the ban is 3 years. Over a year and the ban is ten years. Apply for a job here with an overstay and invalid visa pretty much guarantees being picked up.

Additionally, any illegal presence, including under six months, usually results in a denial for violating your status and makes you a likely intending immigrant.

Illegal status is any time in the U.S. when you violate the terms of your admission or stay beyond the admission date, or anytime after entry without inspection.