I've searched high and low for a decent cheap high deductible health plan in New York (I'm a healthy 27 yr old sole proprieter). For whatever reason, they don't seem to exist. Can I use my mother's Kansas address to get a plan?

You probably can, but the plan probably won't be valid as in-network in New York state, if it's even accepted at all, and then you'll be paying significantly more out of pocket for services anyway. So, that kind of defeats the whole purpose. (And that's only if it's not deemed to be fraudulent, which is a whole other ball of wax.)

Call a local insurance broker and see if they can hook you up with what you're looking for - more often than not, you can get a plan like you describe for $250 a month or so.