Hi @nonibfit!
Are you using the Acuity plugin for Square to sync the two together?
There is some info here which may help: Acuity Scheduling and Square | Square Support Centre - AU (squareup.com)
I don't personally use Acuity (we use Square Appointments), but I would assume the plugin would then bring customer details in.
If you haven't looked at Square Appointments already, for what it's worth I would highly recommend. When we set our business up in 2020, as an existing Squarespace developer, I was already set on using Acuity (because it was built into Squarespace) — I didn't even know Square had a booking system (until we bought the reader to use for Acuity and then discovered how much better Square Appointments was!). And it was free! They have subscriptions now, but in comparison to Acuity from a user experience, Square is miles ahead of Acuity (I manage an Acuity system for another client, and its so cumbersome!).
If you were using Square Appointments, it automatically has email receipts option built in at the end of the transaction / checkout linked to the customer profile.
