If I were a young Major Leaguer, trying to establish myself, I would definitely have a "me" over "team" attitude.
For example, I would much prefer go 3-4 with a couple of RBI's and the team lose, than to go 0-4 and the team win. Because the more I go 0-4 the more likely I'll be on my way out, regardless of how the team fares. But if I gather some decent stats, even on a crappy team, then my chances of sticking and making more money are higher.
While we would like to think all players are concerned about winning, the main thing on every players mind is money. How much they get paid is in direct correlation to their statistics.
Now, if I became an established player, winning would become more important. By then, with the salaries doled out these days, I would already have financial security and would want to play on winning teams while still maintaining those ever important stats and paychecks.
I've been in big league clubhouses of winning and losing teams. While in the winning team's locker room, I like to observe the guys who went 0-4 or maybe a pitcher who gave up some runs. They aren't always happy even though the team won. I've seen guys in the losing locker room who had a few hits and RBI's, maybe a homer or two, and they look like they are sad because the team lost. But I'm guessing on the inside, especially if he's a younger player, that he's happy. Basically he's thinking, "I got my hits. Even though we lost, not my fault."
Bottom line is, players are happy when they do well and if the team wins that's a bonus. If a player has a bad game and the team wins, well how happy can you be if you didn't contribute and your stats take a beating? At least you can pretend to be happy. If a player has a bad game and the team also loses, well then, that's a lousy day.