They don't; but governments aren't bound in the same manner regarding the employment of military force against other nations (and subnational groupings not on American soil) as individual citizens are bound with respect to self-defense and the rights of one's fellow citizens.
It would seem the United States' government might just violate someone else's "quiet enjoyment of liberty and property" if a Trident nuclear sub unleashed its fury on them.
As long as that "somebody else" isn't an American citizen on American soil, it's frickin' irrelevant.
Does the national government not have the right to own nuclear weapons either?
They do. As I observed, the rules for defending yourself are a wee bit different than those for nation-states fighting wars.
Are you a member of the "Campaign for Nuclear Disarmament" or "Greenpeace"?
No.