Keep it vague like you keep your illegal status vague to anyone that asks.

In the workplace it is a liability.

I just say im authorized to work for any employer via email. This way anything is in writing. Any further probing questions from them exposes them to violating workplace laws. I''m not a lawyer though.

I know in some situations they can demand you to be a GC or USC, but I have tentative evidence that its only for security reasons or jobs relating to security for the country.