It is still a question of will power and resources available. If the task of defence is assigned to a few “defence companies”, I would suspect them to switch sides, if the money seems right.
Armed citizens would be a far bigger problem. But even they can possibly moved into submission. With the whole subject, I’m recalling the Anglo-Boer-War or second Freedom-War as we call it. Although the Boer Republics had parliaments for their “Burgers”, They were anarchic in many ways. Most of the White population was armed and minding minding it’s own business (In some areas you had Black chiefs ruling their tribes-some of them got involved in the war later). When the Boer armies were facing trouble, the Boers resorted to guerrila warfare - which made the war a very expensive enterprise. Then the British started the scorched earth policy and put the Boer families (as well as Bantu supporting them) into concentration camps where many died. Still the war continued - But there were those Boers that joined the British cause (to end the war and become a British colony). When everything became seemingly hopeless, also the last Boers resigned from the war and the Boer republics were finally occupied.
This example shows some things:
- A free society can make it very difficult for a state (even if this is far bigger) to allow occupation of it.
- Condition is that the citizens are armed and willing to fight.
- Heterogenity can become a problem for the free society and be instrumentalized by the state forces.
- One problem can be people that only mind their own business, without resisting the invading state forces or caring about the cause.
- The biggest problem maybe joiners to the state cause that even fight even against the other members of that free society.
I might add that some of the mining companies may have even favored the take over by the British - A matter one may have to elaborate on, when analyzing this example.
In that situation things would become very difficult for the US. Just that I do not think that there won’t be nobody in that country to assist them. Obviously there maybe those that think they can gain by collaborating with the invading USA.
Just what would the “private protection agency” do in a situation having nukes at their disposal - Isn’t there the possibility that they will go act like a state.
At the moment I also think of the Condotteri and Landsknechst of the early modern era.