Who is going to seize my property? I'll just have my bodyguard service prevent anyone from stealing my property. Besides, there will be no one to do the seizing because the government will have gone broke by this point. This wasn't possible before because accounts can be frozen & cash can be stolen. With brainwallet, there's no bank account, hidden stash under the mattress, etc. I can take it with me anywhere without worrying about anyone stealing it.
So... the government won't be able to afford collecting taxes from you? How do you expect the government to go broke BEFORE they're rendered unable to collect property taxes?
And what about the public? I doubt they'd just put up with some guy using up space and not offering anything in return. Especially once you fought off the cops, you'd have molotov cocktails flying in your window. Without police protection you'd need to pay the people off, which effectively functions as a tax. Remember, the poor people will have Bitcoin too and it's a lot cheaper to attack than defend.
In any system, statist or anarchist, you simply will never get away with using scarce natural resources for free.