If you can't get in on PS4 and you have a copy for PS3, try logging in on PS3
I launched it on PS4, updated ffxivboot successfully then it went to black screen, timed out after a few minutes and never showed me a login screen. I just tried it on PS3, it's patching now. Perhaps the PS3s use different servers.
Since the PS3 game is considered a separate license from the PS4 copy, you must have purchased a copy on PS3 to play on PS3.
Update: just got in!
https://pbs.twimg.com/media/B5voWZuCAAAEFrf.jpg:large