There is no central server the goverment can shut down and stop it, it's more like torrent traffic but without having to rely on a central website to get your torrents.
You are right but they can try to stop miners from running nodes. They can do so by banning mining all over the world, but they cannot completely stop mining operations since they can opt to mine bitcoins covertly, hiding from the government(but that will basically be rule breaking if they do ban mining all over the world). For now, I think it is less likely because not all governing authorities of every country see bitcoin as bad, some of them see benefits in legalizing it.