I used to work in reality TV (Thank goodness I don't anymore) and I've had friends who have worked on Big Brother so I can actually tell you how it happens:
Is there a script involved? No. The show is streamed 24/7 so the people are who they actually are, for better or worse. This makes this one of the more "real" shows on reality television.
That said, there definitely is producer involvement. The production team has meetings to determine how they want to steer the course of actions via competitions and games that play to certain people's strengths. They also add in moments where America can vote on things at key moments that they think will benefit certain players they find drive viewership.
So in short, there is definitely a hand in things...but if anything it's a hand that comes by using knowledge of the houseguests against them and American viewership's preferences to further their own cause. I WILL say that this leaves a lot of room for surprises and things that are completely out of anyone's hands, which is fun.