I would have asked you to check the questions on SO first, but I think is no way to do it the way you are trying. Well, check these links: how to run code on window close or refresh and how to detect it was close
Basically you can't. This is one of the reasons why many applications show 'last login time' and not logout time to users, see if that works for you.
Regarding saving the logout time of a user: I would say do not store the value at all. The sessions should be allowed to expire based on configuration: users can be logged out even by staying inactive a long time. A logout time for such users may calculated later, based on their last activity time, (which you can store on each page load/call to your server, when you refresh the user's session timer) and the session timeout value. Now this can be calculated when the user logs in the next time, or using a background process.