Did you save the name and password in the Firefox Password Manager?

Tools > Options > Security: Passwords: "Saved Passwords" > "Show Passwords"

See Password manager - Remember, delete, change and import saved passwords in Firefox

• Websites remembering you and automatically log you in is stored in a cookie.
• You need an allow cookie exception (Tools > Options > Privacy > Cookies: Exceptions) to keep that cookie, especially for secure websites and if you let cookies expire when Firefox closes
• Make sure that you do not use Delete browsing, search and download history on Firefox to clear the "Cookies" and the "Site Preferences"

Maybe that website uses autocomplete="off" in their login page script to stop web browsers from offering to save the password for that web site??

Saved Password Editor extension allows you to manually add usernames & passwords to the Firefox Password Manager.
https://addons.mozilla.org/en-US/firefox/addon/60265

The issue that I'm having is that I have to check the, "remember the passwords for sites" box each time I open firefox. The box won't stay checked.

See http://kb.mozillazine.org/Preferences_not_saved

Start Firefox in Diagnose Firefox issues using Troubleshoot Mode to check if one of the add-ons is causing the problem (switch to the DEFAULT theme: Tools > Add-ons > Themes).

• Don't make any changes on the Safe mode start window.

See:

• Troubleshoot extensions, themes and hardware acceleration issues to solve common Firefox problems
• Troubleshoot issues with plugins like Flash or Java to fix common Firefox problems