Visually it's dramatic and maybe "looks similar" to Tibetan monks' self-immolation or some such, but... I'm not sure I see the point of comparison with dictatorships or occupations, exactly.
Obviously the US isn't a dictatorship or commando state to compare with China, or under occupation or heavy foreign domination, like Czechoslovakia in the seventies (there were a couple of instances of self-immolation as protest in Prague, even years after the Warsaw pact had invaded in 1968). But because self-burning is a potential act of fast public suicide, where the protester really stakes his/her life and body on the wish to make a point, and solidly risking death or lifelong disfigurement even if they would be taken to hospital
, there is a very high threshold against attempting it unless one feels the situation looks absolutely desperate. I think that makes it quite different from going out into the street with a placard, posting a Youtube video or arranging a sit-in. It's also typically a solitary act.
Obviously there are quite a few people who want to reduce what they perceive as central government overreach (and maybe a very vocal few who would have been much happier if the South had been able to stay out of the Union)... But the best case of historical comparison in the US might be with the Native Americans -- whose previously separate territories were occupied and sometimes that illegally even under the scope of US law. But, I don't think too many of them are going there. If they were, I can't imagine why they would start over this and not much, much earlier.
There are also many countries where in fact the central government has much more uniform power over the whole country. For small examples, Chinese students are surprised when I tell them each US state government can make its own rules about age of consent or age of driving eligibility. Or you could look at France, the UK or Japan, or several smaller countries (particularly post-colonial ones) and see how much power is still vested in their capitals -- how much of the industry is in that one city, how many people practically have to move to the center to find jobs, how many provinces rely hugely on investment policies determined periodically by the center.
I think there are lots of places where central government is much more influential overall than in the US. Some of those places are much more violent (e.g. contemporary Nigerian history), but many others seem more patient and functional to me than current US politics. If the issue were merely natural resistance to central authority, I would expect the more centralized, more bureaucratic and uniform countries to have stronger chaos in the streets (and I don't mean just a few French farmers blocking the highways in protest now and then). Maybe I'm just not close enough... The British did have some riots not all that long ago, but then so did the US with WTO etc... Still, the Japanese government is structured so that they can redo the leadership every few months if they want, so I'm not surprised if I don't hear much about high-profile demonstrations like this there too often...
Just a quick reply, cause I'm going into town in a while: in most democratic countries parliament can't make the state apparatus ground to a halt sharply just by deadlocking the budget negotiations, whether it's by serious political discussion or by filibustering. In places like the UK or most of Europe, the cabinet will have a majority in parliament on its side by default, so it will be able to anchor the broad outlines of the budget with its own people at least, and hopefully some of it with the opposition too. If it's a minority government, it will have adapted the budget to what is feasible to get through parliament and everyone will have done some preparations and homework before the budget proposal lands on the floor.
If it's not a parliamentary system or not even a democracy, the executive branch in a country would typically be strong enough to keep state and public business going a bit past the date when a new budget should have been voted in. This kind of thing typically only happens in some really poor countries with low state liquidity and low GDP (like in Puerto Rico in 2006, though PR is not a sovereign state). I can kind of see that some Americans would say "it's better to have the state activities fall to the ground sometimes than to risk oppression" - in this case, risk funding of the ACA including abortions being made through public money and public insurance, and made locally in the state - but I doubt that's how most people, as citizens or ordinary workers, emplyees and taxpayers see it. When the central government shuts down with a loud thud, it soon starts to pull everything down with it. Including people's day-to-day lives and routines.
On a plane of political ideas, I think it comes down to whether the state (and the central spheres of authority) would be seen as ultimately just a necessary evil - and a potential fuckup douchebag which should ideally be restricted to voting on the colour of parking meters in the Capitol's car park - or as a way of channelling people's aspirations and taking on challenges that are too big and too costly for any one family or town to (always) take on by itself.