natharward

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

Ok I've been looking into this some - and it looks there are some services you can pay (like $19/mo) to synch your square and squarespace inventory. But they both have public API's (assuming you are on the advanced commerce plan here) so I'm pretty sure my company will just make a custom inventory synchronizer.

As expected, there is quite a lot of confusion about the Point of Sale services provided by Squarespace, especially as they partnered with Square - a major player in the point of sale payments market. The company names are just too similar!
To ensure everyone is on the same page, I'll try to clarify some of the areas where others have been confused.
 
Square isn't Squarespace
Firstly, it's important to remember that Squarespace and Square are completely separate companies. They are not connected, and neither are their ecommerce platforms. You need to select one or the other for your inventory management. You cannot link or sync them directly, although you can pay additional monthly fees for third party 'sync' services.
 
Availability
The options available to you will depend on your location.
Squarespace POS is currently only available in the USA. You cannot use it in Canada (or anywhere else) yet.
Square POS is available in many more countries.
 
Moving from Online to In-Person
If you are based in the USA and have been using a Squarespace Commerce plan to sell online, you can now sell in-person too, using the same Squarespace platform. To do this, you need a newish iPhone, iPad or Android phone that is capable of running a recent version of the Squarespace Commerce App (you'll find specific App version details in this support article). You'll also need to open a Square account for in-person payment processing and purchase a specific type of card reader (you cannot use fancy Square registers).
If you are based outside the USA and are selling online with Squarespace, there is unlikely to be a Squarespace-only option available before 2025. 
 
Moving from In-Person to Online
If you are based in the USA and are using Square for in-person sales then you have two options available to help you move online:
You can build an average Square Online Store and start selling online really quickly. Square will sync inventory between your POS and your Square Online Store if you create it using the "online store" option within your existing Square account. You can build a beautiful Squarespace store and then transition your inventory off Square and onto Squarespace. Once complete, you can use Squarespace for both online and in-person sales. If you are outside the USA and you currently using Square for in-person sales then your options are:
1. You can build an average Square Online Store (as described above) for your online sales.
2. You can build a beautiful new Squarespace store for your online prescene and then purchase a third-party solution to bridge the gap between the two different inventory systems. Bear in mind that there will be additional costs to use a inventory-sync service such as Trunk Inventory or SKU IQ.
I hope this helps.