If the users are using PuTTY on Windows, this isn't really a Linux question. The close button is part of the Windows OS, you're not really going to be able to disable it.
If your users, once logged in, use screen, then connect to the database, their database session will keep running should the user close PuTTY/their computer without logging out. It may be possible to automatically use screen when the user connects. Of course, you must decide whether having users left logged into the database (possibly for several days) if they don't log out properly is better than the current situation.
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