Try trouble shooting this methodically.
Start with a clean slate: Purge your temp internet files and old cookies via your browser tools >options.
Now try logging in and click the remember me box.
If you are still getting logged out at page changes, then you (probably) have a cookie handling issue. This site uses cookie exchange (little bits of data installed on your system by the site) to remember who you are at page changes.
Many internet security programs either block cookie exchange or purge cookies at page changes. You can specifically enable cookie exchange with this site and you may have to do it in two places.
In your operating system settings
In your virus protection program settings.
If you set your OS settings to medium that should be adequate or you can go into the settings and allow this site.
Then repeat process for virus software.