i would also like to report the same issue. I have added https://square-enix.com
though i could not add http://square-enix.com to allowed sites, due to it not being secure.

this issue has been going on since the release of the new client for me, and i have the same exact symptoms.
no add in or antivirus is running when i was troubleshooting this issue.
how does the client save the login name is what i would like to know would help me troubleshoot this issue