Take the new job without the conditions. If you are collecting unemployment benefits, this could create problems with the EDD as to your employment status. Even though you will not be collecting unemployment when you start your new job, the EDD can still look back to see if you would have to repay any money obtained fraudulently. This is not likely, but it is best for you to avoid your old employer. You can be thankful to your old employer for finding you the new position without sacrificing yourself out of a misguided sense of loyalty. It sounds like your old employer thinks it would be to his advantage to have you as an employee in his efforts to merge with the other company. However, he gave up that right when he quit paying you.
Good luck in your new job.
No, your former employer cannot legally do what it is attempting to do. You are entitled to take a new job wherever you wish and to receive the money owed to you from your former employer as well. Of course collecting from a company that ran out of money may be difficult, but given that dynamic, why would you ever agree to continue to get paid from that broke company?
You can proceed against your former company to get the money owed to you by filing a complaint with the DLSE in California. You can do that at no cost to you, so that if the company is truly broke, you will not be out a lot of money.
Good luck to you.
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Based on your facts there is no basis in law for the request of your former employer.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.