Thank you for the update Kluaf.
We have not had any other reports regarding issues with the various Square Enix websites, so I can't say with any certainty that it's the websites themselves. I'm going to assume that it's not just the Square Enix websites you are having trouble with, you just haven't visited others that would exhibit the same symptoms. The commonality between all the devices you've tried seems to be Verizon and the fact that it is a mobile network.
For the white screen you are getting on the PS4, you should try the suggestions listed above (however, some of those are designed with a router in mind, not a mobile hotspot).
We hope we can be of assistance in helping you enjoy the exciting world of Final Fantasy XIV!