Were the East India Company and the Hudson's Bay Company government-run?
Short answerNo. They were private companies, and the East India Company was later taken over by the British government.
NotesNeither the East India Company nor the Hudson's Bay Company began as a government-run enterprise. The East India Company's later takeover by the British government is connected here with failures of company administration. The comparison also distinguishes their territorial control from the influence of Fordlândia and American fruit companies. Private ownership and the exercise of territorial power are separate questions.