Do I have rights to our residence?

You can certainly argue for staying in the house to avoid selling at a loss, but you can't "insist" on it, because ultimately it's up to the court to make the call.  I think if you want to go this route, you'll need a good lawyer, because the arguments that would support your position are pretty involved.

It might not be a bad position to take, though, even if it turns out that you have little hope of getting the court to actually let you stay on in the home.  If your wife wants a short sale now, it's possible that the judge might make her take on a bit more than half of the debt, if you have presented an alternative.