unless you have a lot of games etc registered through it which I doubt since you don't recall it I would just make a new one they do have a account restore for passwords but you will need to remember which security question you picked and the answer.
Not sure about screen names but if you try the password restore the email may have your screen name.
"It is the mark of an educated mind to be able to entertain a thought without accepting it." -Aristotle
The same thing happened to me. I spent over an hour looking before I finally contacted customer support. I talked to someone via online chat and after some questions they gave it to me. It turns out my Square-Enix ID was NOT my user name, but was in the format of three letters and two numbers (Like ABC23), which was my initials and birth year.
I don't remember whether they required me to have my Square-Enix ID in this format or not, but I know that I usually wouldn't choose to do that.
The bear bit off his legs one by one, then pulled him through the snow by his wang. It changed his life and the way he urinates.
Same thing happened to me, I went on my email and added square to my contacts, found it by searching ''From my contacts'' I use this particular email JUST for forums/games etc... so it was pretty easy for me to find. Just check your emails.
We shall heal our wounds, collect our dead and continue fighting. Mao Tse-Tung