There is not a general answer to this question.
Each government organization is generally going to have an "organic statute" that governs it. It will generally direct who can serve on the board of directors. Particular organizations may or may not have a nationality limitation.
No E.U. governing document prohibits non-nationals from serving on the board of a government owned organization, although it would be unusual. Still, one can certainly imagine circumstances where it might make sense.
For example, Angola is a former Portuguese colony. Perhaps Portugal might want to form a government owned corporation to manage assets that remained titled in Portugal or in Portuguese citizens who then transferred their interests to the Portuguese government in Angola at the time of Angolan independence (e.g. mineral rights). It might make sense to have one or more Angolans on the board of directors of such a holding company, even if it was owned by the government of Portugal.
There is nothing on its face that is wrong with doing that legally, although politically, that might not be a popular decision with some Portuguese political factions.