I'm not sure about what the 406 code could be, but SE posted a thread about the 401 errors at launch as it's actually a common problem with no clear solution (hence why support couldn't help you): http://forum.square-enix.com/ffxiv/t...de-401-and-601

A common possibility is it's actually your internet browser causing the problem - often they have addons or extension programs designed for added security that are blocking a certain function of the Mogstation from working when you enter your card details. Trying a browser with such adds ons disabled may help (the fact it keeps throwing up the same error with multiple cards and even Paypal seems to suggest this is the cause).