If you marry a man with no health insurance if he has medical expenses he cannot pay and has no assets could 'they' come after your assets in NY?

That's really a question for a family law attorney. Rather than worry about it - why not just purchase coverage for him.

If you are talking about expenses incurred after the marriage has taken place, then YES, your assets are at risk. Medical expenses are considered a "common" debt of the marriage, making both parties responsible for paying them. In the event of a divorce, the responsibility of common debts is subject to the divorce agreement.