Are you running any third party programs that may be performing some sort of cleaning process on shut down and or reboot?
Do you have your browser configured natively or by third party plugin to clear cache and or delete temporary internet files on closing or possibly from caching username and password credentials in the first place.
Since you distinctly mention "every website", I can only guess that it is a browser issue and something either native to the browser or a product of a browser plugin or something third party like Ccleaner or other utility that is causing the issue.
May I ask.......
Is this a single browser issue, or does it happen in every browser?
Are you only using IE or is Firefox and Chrome (safari, opera), all behaving the same way?