I would call them back and ask to speak with the next level up.
Explain to them that you had previously called, Square admitted the mistake, but would not provide you a corrected form 1099-K.
You should then explain to them that they are required by law to issue a corrected form and if they do not you will notify the IRS of their noncompliance.
This should most likely get the ball moving.
*A reminder that posts in a forum such as this do not constitute tax advice.
Also keep in mind the date of replies, as tax law changes.
