You can do that anywhere in the world that the US does not have a patent agreement with.

Correct. Most of Europe already have that approach.

Would you have to revoke your US citizenship and become a European citizen to do this legally?

I don't think your citizenship matters, it's where you carry out your business. This is probably more difficult to nail down than you think, e.g. if you advertise to US customers, are you actually doing business in the US?

If you're business is incorporated in Europe then you personally have zero liability. Similarly, if your business assets are solely outside of the US governments hands then they cannot do anything to you. You can readily sell your software to the US without any risk of personal liability or corporate liability.

It's not a question of doing business in the US, it's a question of the US' ability to access your companies assets.

Of course if you're an American citizen operating a sole-trader in Europe it is you violating the patent, no company officially exists. You would be fully liable in the US for everything from your house to the current pair of underwear you have on. If a court decided you were liable, you're screwed. If you're incorporated in a foreign country, you'd never be liable and the courts would never be able to argue jurisdiction.

you might end up paying tax in 2 countries though. I think NZ has some tax agreement with the USA but I'm not sure on the finer details.

If you're an American citizen, you'll be paying tax in two countries anyway.