Just found a solution! Perform the following steps:
1.Hit windowskey-s to open the windows search.
2.Type in "Internet Options" and select the top result which opens the Internet Properties window from the control panel.
3.Click the Security tab.
4.Click the "Trusted Sites" zone.
5.Click the "Sites" button.
6.Add both
https://square-enix.com and
http://square-enix.com to the list of trusted sites.
7.Apply/close out of the Internet Properties window.
8.Relauch the launcher and check the "Remember Square Enix ID" button and login.
9.Success!