I answered another question that had a problem with session being killed when the user edited the web.config on a live site. They were tracking users still being logged in with Session variables (dangerous). But came up with a solution (untested solution) that could help people here.
FormsAuthentication allows you to maintain a person being active and logged in indefinitely. But if they become inactive for e.g. 20 mins they will be logged out which is nice. But to have them logged out at the time the close their browser is not possible (wait for it...) as setting the timeout value to 0 would cause them to be constantly logged in then out again.
So solution : at the time you log a person in using FormsAuthentication you could also set a standard session variable cookie that will be deleted when they close their browser. This cookie would have non-identifying non-account related information. Just a simple "loggedIn:yes".
Now all your code would need to have on it's masterpage/materlayout is a high level call in the page cycle or constructor of the page cycle (or even a custom attribute) that would check both cookie and the user identity:
if(!HasLoginCookie() || !System.Web.HttpContext.Current.User.Identity.IsAuthenticated)
{
// redirect user to log in page.
}
Basically if the cookie is removed when the browser is closed, you will redirect the user to the log in page.
Hopefully that helps (and works. As I said untested).