Same problem here. Windows 10 and no response when pressing the 'Log in' button at all.
This was my solution:

It seems to be something with the browser used for the login (wtf? yes!).

Open the following file with your notepad.
C:\Users\[YOU]\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn\FFXIV_BOOT.cfg

Change the following line:
BrowserType 0

Into:
BrowserType 2

Or try 1, 3, etc. For me 2 seems to work.