I found out what causes this. On Windows 7 64-bit (and maybe other versions), if you have IE's security setting for the Internet zone set to high, the login page will fail to load in FFXIV. It doesn't matter what your default browser is, FFXIV always uses IE.
To fix this, you have 2 options:
1)
Add the square enix sites to the trusted site zone
To do this, open IE and double click on the status bar on the bottom that should say something like "Internet | Protected Mode: On". This will bring up a dialog title Internet security. Click on the trusted sites zone. Make sure the slider is set to medium-high or lower and click Apply if you needed to change it. Click on the "Sites" button. A dialog will pop up titled "Trusted Sites". Uncheck the "Require server verification (https

for all sites in this zone" check box. Edit the entry field labeled "Add this website to the zone" to say "square-enix.com" then click on the add button. It should show up in the list below.
2)
Change the security setting for the Internet zone to medium-high or lower (not recommended)
To do this, open IE and double click on the status bar on the bottom that should say something like "Internet | Protected Mode: On". This will bring up a dialog title Internet security. Click on the internet zone and change the setting to medium-high or lower on the slider and click OK or Apply.
That should make the login screen show up if you had the same problem I did.