Six is a tender age to begin with. I would probably take a common sense approach with him.
First I would simply explain that chess is a game and like every other game, there is a winner and a loser. Not everyone wins 100% of the time and not everyone loses 100% of the time.
When he is playing and he starts losing and wants to quit, I would ask him how it would make him feel if no one else ever wanted to play chess against him if they knew they couldn't win.
If he can't make the natural connection to the example and his own attitude, I would make the connection for him by explaining that how he would feel is how other people feel when he won't finish the game to completion because he fears he might not win (or he foresees he might not win).
I would tell him that chess is like any other academic topic he is interested in and that he will learn a lot by playing against people who have higher skill sets and can beat him.
Sure winning is fun, but losing can be viewed as an opportunity to get better, rather then a negative experience.
I think sportsmanship has to be taught to every child, it isn't usually a natural personality trait. The perfectionism in gifted kids just make situations such as this a little bit more sensitive then normal because you want to teach them to be good sports without turning off their love for an activity.
I think in time he will probably understand and accept what you are trying to convey.
