blocking ffxiv doesnt reset a router , so the problem is the router itself.
Login to your router and check the settings /allowances .
I have had something simular happen , the router set on high security let only so much through at a time and then it blocks.
It was a build in firewall that most routers have " when they feel they get too much info at a time " they think thier build in firewall is bombarded AK attacked and automaticaly block all in and outgoing connection.
Happens to me at times when i download programs on Utorrent , because of the high download speeds my router will shut itself down someties