Working on non E-verify company during OPT extension period. Will it create problem if my company file for H1B this year?

No, working for a non eVerify employer is not a problem. At this point, you should probably remain in the U.S. until you find an employer willing to file for an H1-b visa and you actually have been approved for an H1-b visa. In the past, the USCIS runs out of H1-b visas rather quickly when the USCIS allows H1-b filing in April. It is unusual in such situations to get an extension on OPT unless there are other facts still undisclosed.

This means that if your newer employer is willing to file an H1-b visa petition, you should and your employer should contact an experienced immigration and visa attorney who can help you with this situation.

If aware, your school may take action on your F-1 visa, which can complicate matters. Before you report any change in employment, you should discuss this matter with experienced immigration counsel. Your F-1 visa status should remain undisturbed. At times, there is confusion among International Student Officials, when students change jobs. By law, if you have a position that relates to your degree, then you should be allowed to continue and change status to H1-b unless action is taken in SEVIS or you leave the U.S.

The above is general information and does not create an attorney-client relationship.