The response to play lots of board games with him is great; be sure to give some praise for his good play -- "Good move! " or "That was a smart move there!" etc. And model good winning and good losing yourself (and talk to your teenage babysitter about always doing the same), by being modest but happy when you win and saying "Good game, you played really well" to the winner when you lose. Of course you'd do that anyway, but make sure he hears it coming from you and especially the babysitter -- I'm betting a six-year-old boy just idolizes his teenage sitter, so the sitter's input could be very helpful here.
I know families where kids have to shake each other's hands and say "Good game" when they play a board game or organized game, even if one of the kids is steaming.
Also, I would end games immediately if he gets angry, especially if he can't calm down after a warning. (And again, have the sitter do the same things you do, the same way - keep it consistent no matter who's in charge.) After a warning, if he gets angry, lashes out verbally, throws a game piece, whatever, very calmly say, "I'm sorry, but we can't continue to play because you're choosing this behavior." And start putting the game away. Again, I'd be sure he got a clear warning first that a second time means the game ends right away.
I think this is pretty typical behavior in many kids in early years, and good for you for wanting to teach good sportsmanship now, rather than having an older elementary kid on your hands who is not chosen for games at school because of his attitude, or who is always in trouble on sports teams because he mouths off at other players or coaches. You can do this!