I'm also having trouble to queue up.
Once enqueued, when you cancel out, you can't queue up again and have to relog.
Otherwise, connection can't be established.

I can confirm though, that only some players are affected. My girlfriend's account does not have any problems.
We're using the same internet access and even tried to group up and enlist together, which results in her being able to enqueue, and I'm getting the errors statet above.

Need some quick fix, SquareEnix!

Cheers.