Mailchimp and Klaviyo both have real, built Squarespace extensions covering email sync and basic customer data, worth using before anything custom. Beyond those two, Squarespace's own extension list thins out fast.
Salesforce, Pipedrive, ActiveCampaign, Constant Contact, Trello, and Slack all show up as common 'does X integrate with Squarespace' questions, and the honest answer for each is: not natively. Zapier and Make both support Squarespace as a trigger source and can move basic data, a new order, a new form submission, into any of those tools without custom code.
Payments run through Squarespace Payments or Stripe depending on your region. Square is a separate, unrelated company despite the similar name, so there's no built-in bridge between a Square POS and a Squarespace store beyond both accepting card payments independently.