The "add service account" button only appears once you have at least one valid service. Right now you have a trial account which doesn't count (bad website design?).

So like the above said, you need to create a new Square Enix account, but not quite for the reason they said (to the best of my knowledge you can have multiple Steam service accounts on the same Square Enix account, but I only have 1 Steam and 3 non-Steam on mine so I can't confirm that for sure, but there shouldn't be any reason why not as they're all separate accounts within the same parent account).