What was the most important rule at Lark Creek Elementary?

Chapter 7 of Bridge to Terabithia introduces the reader to the most important rule at Lark Creek Elementary—family came first and family did not get discussed at school.

There was a rule at Lark Creek, more important than anything Mr. Turner made up and fussed about. That was the rule that you never mixed up troubles at home with life at school. When parents were poor or ignorant or mean, or even just didn't believe in having a TV set, it was up to their kids to protect them. By tomorrow every kid and teacher in Lark Creek Elementary would be talking in half snickers about Janice Avery's daddy. It didn't matter if their own fathers were in the state hospital or the federal prison, they hadn't betrayed theirs, and Janice had.

Janice Avery's father beat her. The abuse allegations were incredibly upsetting to Leslie and Jess, but Janice's friends Bobby Sue and Wilma had taken advantage of the situation and blabbed to the whole school after Janie confided in them. By the end of the school day, Janice was mortified to think that people would be laughing at her as she passed them in the hall. By sharing her private information with a couple of close friends, she'd opened herself and her entire family up to ridicule.
Leslie suggests that Janice pretend she has no idea what Wilma and Bobby Sue are talking about in hopes that the vague denial will make the story less interesting and quickly forgotten.

The answer to this question can be found in chapter 7. During the second half of the chapter, Jess and Leslie discover that Janice Avery is having a hard time with something. Jess tells Leslie that they need to find out what it is and try to make it better. Leslie discovers that Janice's father beats her badly on a fairly frequent basis. On this particular day, Janice confides in her friends, and her friends blab about it all over school. Jess is horrified about the entire situation because it is a clear violation of the most important rule of Lark Creek School. The rule is that what happens at home, stays at home.

That was the rule that you never mixed up troubles at home with life at school. When parents were poor or ignorant or mean, or even just didn't believe in having a TV set, it was up to their kids to protect them.

Jess explains to readers that no matter what happens, a person should never betray their own family. Janice Avery betrays her family by confiding in her friends.