As I see this is your first post let me welcome you to the Square forum.
"The app is made by Square and published to QBO app store, and made available to QBO subscribers."
As I have see posts from the past years about problems getting data from Square to Quickbooks (Online and Desktop), I would like to know how you know Square made the app for transfering data from Square to QBO? I followed your link and watched the video that was in the link, it was actually uploaded by Quickbooks Intuit not Square. If you go to the Square app and Integrations page the QBO integration only gets 3.1 stars out of 5. https://squareup.com/us/en/app-marketplace <--- link to apps that integrate with Square.
I do also remeber finger pointing between Square and QuickBooks saying it was the others fault for why things do not import properly. So here is my thinking Square does not really have Classes of Items for different Accounting reasons like QuickBooks does. So QuickBooks Says its Squares Fault for not having these different Classes in the Square APP. The Square APP being the POS apps, not the integration app. If that makes sense. What you could try doing is having the import to Quickbooks look at the SKU of the item and have it pull the Item from Quickbooks Items to fill the fields.
"The app matches Square's item names to QBO's item names. The QBO item names are in turn specifying an income account and a class. When processing happens, income account is set successfully in QBO, but class is not."
Once the integration app sends an item from Square that matches the Item in Quickbooks, Quickbooks should be the one assigning the class as defined in Quickbooks. Square is just reporting which Item was sold. Then QBO says oh the Red marble was sold lets find out what Income account that belongs to and which Class of items that is. Since Quickbooks finds the Item and the Income account why is it not finding the class for the item? All QBO needs really is the Item (SKU), Quanty sold, Date of sale and The price showing discounts if applicable. Using this information QBO should pull all other class and information about the item like Cost when bought, how many are in inventory, The Class of the item etc.
Now if you look at my Quickbooks DESKTOP items fields for this School bus Cookie Jar, I have no field to define a Class, But I can assign which Asset Account QBD uses, The Cost of Goods Sold Account to be used and the Income account. You may have to look at your Items in QBO to define all this correctly. I chose this item because I knew we were out of them. 🙂
"Upon contacting QBO support, I was informed it was the fault of the Square app for not properly specifying a class."
This statement goes back to the fingerpointing I referred to earlier. Then you continue on and say "The app matches Square's item names to QBO's item names. The QBO item names are in turn specifying an income account and a class." Which I take as The Item name from Square is Matched to an Item name in QBO which then would pull the Income and Class account information. There by Square not needing to define a CLass since once the item names match QBO item names are in turn specifying the Income Account and Class. Square also does not specify an Income account in there data, so how does QBO know this information?
"Where would I need to turn to in order for someone in Square to confirm their app in QBO is functioning properly and makes proper API calls to QBO?" I would call QBO support and ask them how they get the Income Account when Square does not specify that either. Shouldn't the Class acoount be gotten from Quickbooks Items account the sameway?
Sorry I have a Bitter taste in my mouth about QB's in general after being lied to in 2020 when I called them about clarification on the 2021 desktop verisoin and if it was lifetime or subscription, they lied and I fell for it and got the subscription which I did not want and am now back on my 2020, lifetime subscription and no more yearly fee to Intuit.
