To reiterate what Paul said, looks like the Squarespace <> Square integration is merely re: hardware. There's no software or systems integration.
See attached image, pulled from https://www.squarespace.com/ecommerce/point-of-sale
If you want a webstore that syncs with your POS for a physical store, you EITHER commit to Squarespace OR you commit to Square ... at least for inventory mgmt and payments.
SS solution: use Squarespace commerce functions for online sales + use Squarespace APP on an iPhone iPad etc WITH a Square card reader for brick & mortar (what I'm not sure about is whether this routes $ into your Square account, which is not the same as sending money into your Stripe account via SS Commerce on your webstore)
Square solution: Square POS terminal for brick & mortar + Square Online Store for online sales ... do NOT use Squarespace Commerce functions
Seems like ... best thing to do is go with Square. And then if you need a website more robust than the Square Online Store, have a Squarespace site without Commerce, and put your Square Online Store on https://store.yourdomain.com
