my plugin check takes me to wrong page "https://www.mozilla.com/en/404", FF12. How do I fix it ?

The problem is on the Mozilla servers, in particular how it handles redirects. Mozilla are aware of the issue and should fix it soon.

Pending this, you can use this link to go to the plugin check page - https://www.mozilla.org/en-US/plugincheck/

Thanks,

My Wife's laptop works correctly though. Why is that ?

This has been fixed on Mozilla's side in Bug# 752232

It tries to use https://www.mozilla.com/plugincheck/ and then gave a error instead when redirecting to https://www.mozilla.org/plugincheck/

Many Mozilla pages that were on mozilla.com have since been either changed to or is being redirected to mozilla.org and sometimes issues have arisen.