We've only had the one episode of headlice, and they don't seem to be an issue where we live now (no rules of having your hair up etc).
It's not so much that headlice "like" clean hair, more that they are not a consequence of dirty hair. There used to be a very strong view that really only poor children had nits and it was because they didn't wash. So when children did get headlice it was felt to be pretty much their own fault, and a fair amount of bullying went on. This is probably when the OP's acquaintance was coming from.
Now there is a correlation between children who are neglected and headlice, because their parents simply aren't going to bother with the hassle of treatment, and may well not even notice that they have headlice, or care that they are scratching. It is unlikely that this group of children are highly represented at private schools (but then they are also the children most likely to miss out on school full stop).