Hey there! You might be able to use Square’s API to do the second part of this using Zapier’s webhook task:
You’d want to use this API endpoint in Square:
https://developer.squareup.com/docs/orders-api/pay-for-orders
(And you can use Zapier’s Webhook for this: https://zapier.com/page/webhooks/)
As for the trigger - you could perhaps use Acuity’s webhooks to achieve this: https://developers.acuityscheduling.com/docs/webhooks
I’m really sorry that this is so technical - But aside from asking the developers to create these triggers/actions - You will need to use these more technical tools.
In short - The good news is that you’re trying to do definitely looks feasible.