Solution! If you have a Squarespace Commerce plan, you can set up the free TaxJar extension. This is now calculating State and County taxes for our products, and it allows us to add a tax category on individual products that should not have taxes! In our case we are using tax category 'General Service' for non-taxable products, and we did not enter a category for any other products. We have all of our products set up as Service products so our customers don't see any shipping options, as all of our products are picked up in our studio.
For taxable service products, it is not using the billing address of the customer to calculate tax, but seems to be using the Business Address set up in Squarespace Business Information. It does not seem to be using the address set up in Tax Jar Profile either (That won't even let me enter our zip code for some reason!)
This is working perfectly for us!
We did have the option to tax Service Products set up when we were doing manual taxes. I don't know if this still needs to be set for TaxJar. I just left the manual settings alone after setting up Tax Jar.
One kind of weird thing is that on the customer confirmation email, the tax is just shown as 'Tax' like it did when we were doing manual tax. But on the 'New Order' email to the business, it is broken down by State and County taxes. It also has these columns when you download order information from Squarespace.