That's just something someone wrote on Wikipedia. Do you think that if the liberland people showed the Wikipedia page to the police, they would have backed off? The only logical conclusion here is simply that the Wikipedia article is wrong.
They experimented, and they discovered what happens if you try to claim land that isn't yours. They also discovered what happens in a libertarian world when someone stronger than you wants your stuff.
What will happen is much simpler. They won't give you the room key if you don't pay, and will eventually ask you to leave the premises if you're not a customer.
Does Costco do "business"? It buys merchandise from suppliers, sells to people. It's incorporated. Its shares are traded on the NASDAQ. Yes, evidently, it partakes in business.
Is it owned by the public (the state), or by private persons? Private persons.