Integrating your public booking page with Squarespace is straightforward:
First read this article on how to take bookings directly from your business website; copy the embed code
Use the embed code on Squarespace in the manner described on their website here
You may have to disable "Ajax Loading" on your Squarespace site for the Bookem embed to load. You can do so by:
Open a Blog Page then, in the Home Menu, click Design, and then click Style Editor.
Scroll down to Site: Loading.
Uncheck Enable Ajax Loading.