Sounds like you want something very firm. I've found that's good for the back as well, though it can sometimes take a bit to get used to. If you're already on the floor, though, then you must be okay with that :)

Foam core futon mattresses tend to be much firmer than traditional spring mattresses. You can try going to a futon store, where they'll also have frames intended for such mattresses. Talk to the staff there about a trial period and return policy; every place will be different but you should be able to find someone that will be willing to work with you.

Best of luck!

For a bad back, if you were to rate mattresses on a scale of 1 to 10, 1 being softest and 10 being firmest, go for a 7 or 8.