Try this
Theres certianly nothing wrong with the Windows Live servers, they don't save the passwords, your browser does, have you tried re-installing Firefox, this usually fixes the problem, if you can't or don't want to try going to 'Tools' > 'Prefrences' >'Security' Tab > Saved passwords > and then choose remove all, re-enter your passwords that you want the computer to remember and see if it remembers!
Hope this Helps, Jack