No, Square cannot take your Squarespace checkout
That is the answer to the question most people are actually asking, and it is not the one we solve.
Squarespace does not support Square as an online payment processor. Its own help pages say so, its forum says so, and the only checkout options are Squarespace Payments, Stripe and PayPal. Square connects for in-person selling through the Squarespace app and a Square Reader, and that is where it ends. Nothing we build changes any of this, because it is a decision Squarespace has made about its own checkout.
So if you came here to route your website’s card payments through the Square account you already use at the counter, the honest answer is that you cannot, and you should plan around Squarespace Payments or Stripe instead.
The question underneath it is usually a different one, though. People asking this generally have a Square till and a Squarespace shop selling the same things, and what they actually want is for the two to stop disagreeing about how many are left. That part is solvable, and it is what the rest of this page is about.