I've got several diaries - one is like a proper diary, which I write personal things in, another is more of an account of special events etc, and I also have an online diary - but it's only open to people who I trust. Even so, there's obviously things I wouldn't write in the online diary. I'd hate the thought of anyone reading my personal diary, because it's like an unwritten rule that you just don't do that. It's like going through someone's bedroom: OK, so not everything in there's going to be "private" as such, but it's yours and it's personal. So I don't agree with what your boyfriend said at all.
Like some other people said, diary-writing is best done whenever you feel like it or else it becomes a chore. I think it's well worth keeping one, especially as a teenager - I've kept diaries for years and it's funny to read old diary entries because it almost sounds like another person. You can change so much over a few years and it's really interesting to keep a record of it. Also, there's little everyday things that happen and I just think it'd be horrible to forget memories and such...