As for Spooner: I am not saying I agree with his analysis on everything (same for Bastiat as far as that goes). I'm just pointing out that those two represent some useful sources of thought on the illegitimacy of the State. That was intended mainly for people who weren't already involved in studying this particular issue.
The question of "what is property" is an interesting and fundamental one for sure. I'm still chewing on that, honestly. Some days I think I know, other days I'm convinced I was wrong before. Lather, rinse, repeat.
If you have to axiomatically state that private property is good and government is bad, then I don't really see why anyone would have to follow your arguments, because not only are such axioms ridiculous, as seen by the attempt to create a rationalization for it. If the argument according to which the state is illegitimate has as a consequence that so is private property, but you still axiomatically state that private property is legitimate, you've also invalidated the original argument that the state is illegitimate, and now your entire worldview is based on two absurd axioms, and not any logic.
A moral philosophy or political economy is only ever valid and worthwhile if it is based on axioms/values that the rest of society share. For example - negative freedom is good, positive freedom is good, prosperity is good, and so on, which you'll find that people tend to agree with inherently. I don't think that there's anyone for which "private property is good" and "the state is bad" is an axiom.
Also, Lysander Spooner literally is against private property. He was part of the Socialist Internationale. The validity of his rejection of the state on basis of unjustified hierarchy also led to his rejection of capitalism on grounds of unjustified hierarchy. Instead, he was a mutualist. You know, the political economy for which one of it's foundational texts is What is property, which coined the slogan : "La propriété, c'est le vol", by Proudhon.
Bastiat attempted to use this essentially anarchist analysis of the state, but exempt private property from it by inventing some weird standard of government action that has no deeper justification and is pretty much axiomatic.
This is a huge simplification, but if you want more detail you can read "Why Anarcho-capitalism is not Anarchism" either on the Anarchist Library or, if you use Debian, via apt-get install anarchism.