Any student that has graduated would be okay but any student that still is in the school system should not be on
your facebook. My district has warned its teachers not to have facebook pages because of problems it can create.
My husband would be livid (and I would agree) if his district tried to tell him not to have a FB page.
Warn him (or his less tech-savvy colleagues, at least) about some of the dangers and what NOT to post-no problem.
Rules about whether or not a teacher can friend current students-no problem, depending on the rules.
But teachers do get to have a personal life, and telling teachers that they should not have a FB page is way, way over the line.
As far as OPs situation, DH would have no difficulties saying no to a friend request from a troublemaker. He would have done the best he could for that kid when he had him/her in class. After graduation, he no longer has to accept responsibility for that kid....and he would not be willing to have such a brat on his FB page.
He does generally accept requests from former students after they graduate (he teaches juniors, mostly, so we are talking about college students). He gets students (every year) who come back and ask for help with some of their college level classes-and he is usually happy to give a few pointers.
But he has turned down a few former kids-and for similar reasons.
As long as you did the best you could for the student when he/she was in your classroom, don't feel like you have to say yes out of an obligation that, in all reality, expired on graduation day.