I was in similar situation couple of years ago when I was in OPT. I started working with them and later requested them to e-verify their company. My HR lady sat one morning and finished it in an hour. It used to be easy process with questionnaires and couple of forms. Don't know if it's the same now.
It is good for company itself to be e-verified.
May be your company doesn't know pros and cons of e-verification. May be they don't even want to understand.
Either way I wish you good luck in convincing them.