Not Square-Enix support, but...

I've seen ATI's Catalyst Control Center produce this error under certain situations if CCC, the actual underlying drivers, and the version of .NET installed on Windows don't all match. It will not necessarily appear globally, it will depend on how an application is initializing various things. I'd be a little surprised if that causes a crash in the Launcher rather than once the game starts, but... well, I notice the system report has a Radeon card. So I'd rip out all the Radeon software (uninstall both CCC and the Radeon drivers, reverting the system to default graphics drivers), reboot, reinstall the unified CCC/drivers package clean, reboot again, and try once more.

If that doesn't work and a complete clean reinstall of the system hasn't yet been attempted (which I would understand, since that's a nuclear option)... admittedly, a clean install of the entire system might be a viable next step.