Could it be possible that when people close the window when still logged on, that they are automatically logged out.

Ick. If implemented, could such a thing be part of a user's individual options, rather than a general board function?

I think you'll get that behavior already if you don't click the checkbox that says "remember me" or whatever when you login. You'll have to restart your browser entirely and not just the window that you're using, but it should do what you want.

I haven't looked into this particular forum, but the problem will be that the "remember me" sets a cookie on the computer. When you close the window, it won't have a chance to delete the cookie.

As Basher says, don't check the "remember me" option. This should tell the thing not to set the cookie.

Well, it'll still set a cookie, but it'll set a session cookie instead of a persistent cookie. Session cookies only live for the lifetime of the browser they're created in, so restarting the entire browser should take care of it.