SQEX TOKEN should replicate Google 2-step encyption.

[worldofneil]

It could be implemented using IP, so while your modem doesn't reset or if you're using a static IP you won't require to input the code again.

True and to be fair that is how Guild Wars 2 does it. I have to wonder how many people forget they have a security token added (and lose/change their phone etc) if they use it so infrequently.

The launcher uses an internet explorer frame, so they can probably its cookies.

Yes that's true, I didn't think of that. I had a look and sure enough there's a cookie for ffxiv-login.square-enix.com.