Back during the strike in the mid-80s, the NFL had games with scab players. While most teams encouraged their players to cross the picket line, the Eagles coach Buddy Ryan told them to act as a team. So, while the Cowboys had their regular players, the Eagles had a bunch of nobodies. Not only did the Cowboys expectedly dominate the game, Tom Landry put his starters back in late in the game so they could score some more. That was classless.
Thankfully, the Eagles got their revenge when they got their players back. They were winning the game and Randall Cunningham was starting to kneel the ball to run out the clock. Then, instead of kneeling one last time to end it, he fake kneeled and threw for a touchdown. I guess it's little moments like that are the reason that I feel so strongly about the Cowboys rivalry. To me, they are the epitome of arrogance and every little thing against them is a victory of David over Goliath.
Aw, but classlessness goes both ways. I've never seen a ruder group of fans than those of the Eagles (not you, of course

As for the title "America's Team", many Cowboys fans hate it, and it just one more thing to add to why people hate the team. I say let someone else have it. It's nothing but trouble.