I realize this is an old thread, but I thought it was interesting and deserved a bump.
There's many ways Predators could influence future Predator/AVP games, but I'm especially interested in how it'd affect AvP games.
A Predators-influenced AvP game could be set on a game preserve, and the Alien/human campaigns could have you fight both the classic Predators and the black ones. There could be infighting between Predators in the Predator one, maybe even a separate one for both breeds, each with different gameplay. The classic campaign would be precisely that, classic (as in more like the previous games') while the black-centric one could be squad-based as the OP said, with the hounds and the falcons at your disposal. Some missions in the BP campaign could require you to play as the hounds and/or the falcon.
The humans in the human campaign wouldn't have to be marines, nor would the story need to take place in the future, and you could throw in the River Ghosts (stickmen) as foes and even a playable multiplayer class. That, and a River Ghost campaign.
I definitely hope future Predator/AvP games borrow ideas from Predators.