Can I accept CC Payments without using PayPal?

If your purpose is to collect credit card details, I'm sorry, but we do not allow such. It is against our Terms of Use to collect credit card information from your users. We have other credit card payment processor. We have Stripe, Authorize.net, Square to name a few. 

How can we add a spot on our form to collect "square" payments to our Square account, please? 

In the form builder, you need to go to the Payments tab and click on Square option. Please check this guide for the steps: https://www.jotform.com/help/459-How-to-Integrate-Form-with-Square.