STUDENTS at White Plains High School were thrilled Monday when Jodie Foster began filming “The Beaver” there, especially when she asked if some of them could be extras in the dark comedy. Foster is directing and co-starring as the wife of Mel Gibson, a depressed man who finds solace in wearing a beaver hand puppet. Anton Yelchin (“Star Trek,” “Terminator: Salvation”) plays their son, who urges his mom to get a divorce. Jennifer Lawrence (“The Burning Plan”) plays Yelchin’s girlfriend. One scene shot at the school had Yelchin and Lawrence having an argument in the cafeteria. Students were told not to bother the stars, but moms were allowed to watch their kids’ acting debuts. And they all got to sample the lavish “Craft Services” buffet provided on the set.