No. You cannot integrate another payment processor with Squarespace. It’s simply not possible because we only Squarespace themselves have access to the backend code. If you want to use the commerce features that are built into Squarespace you must use the payment processors that Squarespace provides (the recommended route).
The only way to use a third party payments system is to ignore the commerce features in Squarespace and use the third party payments system/store to manage your stock, your orders and your payments. There are many to choose from, for example SendOwl. Also, Shopify offers more than 100 different payment providers. By adding Shopify buttons to Squarespace, you can use the Shopify checkout and payment gateways. You can read how to do this and get a 14-day free trial of Shopify here.