This question doesn't have an easy answer. When you say "refinance", what exactly are you talking about? From context, I think you mean that the two of you would execute a deed to just you. You can easily run afoul of the fraudulent conveyance laws when you do this. Such laws allow creditors to set aside transactions that are done with the actual intention to hinder creditors or that are done for inadequate consideration at a time when the transferor is insolvent. It sounds like both of the predicates for the law would be present in your case.
The bankruptcy code has its own fraudulent conveyance section (548), which includes a lookback period of one year. The trustee can also often take advantage of the state law of fraudulent conveyances, which may have a longer lookback period. There's even an argument that the trustee can use the common law of fraudulent conveyances, where the lookback period is essentially unlimited.
Your situation isn't necessarily hopeless, though. If your equity in the home is less than the amount you can exempt, H's creditors won't be able to get to it anyway. You'll need to ask a local attorney about homestead exemptions to know the answer to that.