The school has no right to step into the home and take over the role of parenting. Moms and Dads should be more concerned about what their kids are posting online than school administrators. However, there is one instance where schools should punish for online behavior. If the false quotes are about the teachers and the staff, the school has a right to respond.
School authority is only for things that take place on school grounds or at school sponsored activities. Home use of Facebook, regardless of the reason or what is published, has nothing to do with the school and should not affect a student's education or open them to discipline. If serious issues are brought to light, it is the parent's responsibility to address home behavior and internet use.