To qualify for unemployment benefits, you cannot refuse work, unless you can prove that conditions are so intolerable you cannot work there.
The information provided above is for general purposes only. It does not constitute legal advice or create an attorney-client relationship. Seek competent legal representation, because the facts of each case are different.
1. You never have to accept work off the books. Work off the books is illegal work.
2. If the work is legitimate, then it is likely that if you refuse it you may lose your unemployment benefits.
3. If you were experiencing a hostile work environment, as in an environment in which you experienced either discrimination, sexual harassment or retaliation, then you should be sitting down with an experienced employment attorney for a free consultation immediately.
This answer does not constitute legal advice and you should contact an attorney to confirm or research further any statements made in this answer. Any statements of fact or law I have made in this answer pertain solely to New York State and should not be relied upon in any way in any other jurisdiction.