The Square module requires that you install all the files in the prescribed folders as per the documentation. If you don't, then the javascript which interacts with the Square servers can't obtain the token/nonce code that is used for processing a payment. When the token/nonce cannot be obtained, the SQ-NONCE-FAILURE error appears.
The javascript hooks into form elements built-in to the standard Zen Cart checkout flow, so those must be present in your template in order for it to work.
Additionally, your OAuth setup must also be completed as per the setup instructions in the documentation.

I just did a fresh setup of Zen Cart v1.5.5e on a brand new site, added the Square module and followed the instructions step by step. The first attempt at payment was successful.

https://www.zen-cart.com/content.php...quare-Payments