It took some moments but I have registered to attempt assisting! Because I juuust got this stoopid bug yesterday. Who knew I had to click the tiny star on my character to select it instead of a confirm character button... anyways!

I believe after some testing most extensive that this is a udp forwarding issue with Square's launcher claiming udp ports for your local ip within the router but never releasing them.

Restarting the pc did not fix.
Purging all forms of temporary cache files did not fix within Internet Explorer (because apparently it uses IE?).
Clearing the temp windows folder was a big ol failure.

However! Restarting my router relieved me of my endless no play button issue and inability to log in using Exitlag. Most amusing is I checked the udp forwarding feature of my router and nadda. There were no claimed ports. What ever Square changed with the launcher is clearly effecting others now though if past experience means anything it will not be fixed in timely way.

I hope this actually works and was not false equivocation. Do update this if it does! I wish to know in case it occurs again. Right after my pre-order too..