After reading the post, USC associate athletic director J.K. McKay showed me the letter below that "Duke" wrote to his dad, John McKay, after USC beat Texas in Austin in 1966.
Wayne was on the sidelines for the game since he was just across the border filming War Wagon in Mexico.
Here is the pregame story as told by Nick Pappas, former director of Trojans' Athletic Support Groups, in Ken Rappoport's book The Trojans: A Story of Southern California Football.
"The kids are all assembled in the locker room at 10 in the morning, and in walks Wayne. Damn, he was fantastic. He walks in with this white 20,000-gallon cowboy hat and black suit - he looked just beautiful. The kids look up, and their eyeballs pop. Here's the REAL John Wayne. And Wayne walks over to the coach and gives him a big hello and squeezes him - you'd think he and McKay were long lost buddies. They had never met before.
"It was beautiful. A former player and all, Wayne gives one of the greatest fight talks you've ever heard - and the kids got all fired up. We win the ballgame 10-6, and back in the locker room after the game, McKay says, 'Hey, guys, how about it? Let's give the game ball to John Wayne.'
"For a moment Wayne stands there - nonplussed. It was probably the first time in his life that he couldn't think of anything to say. Then he looks at the ball for a minute and pumps it like a quarterback. Then he puts the ball under his arm, and the kids break into a cheer, 'Hooray, Hooray.' All the guys joined in. He's still a Trojan."