Websites remembering you and automatically log you in is stored in a cookie that you must allow: Tools > Options > Privacy > Cookies: "Exceptions"

Make sure that you do not use Clear Recent History to clear the "Cookies" and the "Site Preferences" "Browser History"

On the mac that's Firefox>Preferences>Security>(Passwords:Remember passwords for sites)Exceptions. There is no exception for UPS.com Firefox(or UPS) remembers the user name but will not fill in the password. There is no entry for UPS in the saved passwords list. The box on the ups log in page for remember me is usually checked, that brings up the user name but not the PW. If i uncheck the box no user name shows until I start to type. I have lots of other sites that Firefox supplies the PW for. A while ago I erased the UPS.com cookie but Firefox never asked if I wanted to remember the next time I logged in.