Is it legal for a government/company to own a patent?
In the U.S., it is absolutely legal for a company to own and enforce a patent. Every invention must, of course, have inventors (which must be real persons).
According to the following page, the U.S. government owned 47,220 patents between 1969 and 2008, most assigned to the U.S. Navy:
I won't be the one to say this is all illegal.
Does any regular US citizen have the rights to any patent held by their government?
I doubt it. Same as you can't just take up residency in a building owned by the government. Of course, if the U.S. government is able to collect some royalties from commercial uses of their patents, then in theory taxes could be a tiny bit lower without running a deficit. So, regular US citizens do benefit indirectly.