Do I have to work for the company that sponsors me for a green card during the green card process?

You do not have to work for the company that is sponsoring you for a green card during the process.  During the process, you can remain abroad or be in the U.S. in any lawful status.  The key here is that you never lose legal immigration status so that you can apply for residency within the U.S.  If you decide to wait for your residency abroad, you can continue to travel to the U.S. for short stays during the process.

If you are applying for a labor certification (PERM) process, you may be waiting between 2 and 3 years for a green card. If you are applying through the National Interest Waiver (EB-2), you may also be