Using PayPal instead of Stripe for e-Commerce on your Squarespace website
Stripe is the only payment gateway that is fully integrated with Squarespace Commerce, but if you are dead-set against using Stripe (or it's not available in your country), then you can use any third-party payment processing service that supplies buttons or embed code.
Here's a simple example of how you can do this with PayPal.
IMPORTANT: Unlike the Products Page example in the VIEW DEMOS folder, this page is not in test mode, it's live and active. The button is connected to my PayPal account, so clicking on the Pay Now button will take you to PayPal to complete a real payment.
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The button above has been inserted using a Code Block.
The example above uses the default PayPal button style, but you can customize the button's appearance if you know how. If you're not afraid of tinkering with code, then try this site for some simple CSS and this one if you'd like to have a custom payment selector dropdown, too.
How to make it more shop-like
If you'd like a page per product, and you'd like to display all your products in a style more similar to the Products Page that you get with proper Squarespace Commerce, then I'd suggest using an Index grid for your main shop. Then you can have a subpage per product. Try Avenue, Flatiron, Montauk or any other template with an Index grid that allows you to use standard pages within the Index.