Onsite registration

does this have the capability to run on an app on a device like iPhone using Square as the processor and authorize.net as the gateway?  

Our forms are mobile responsive, if you want to embed the form in a mobile app, you can try embedding your form's Iframe code: https://www.jotform.com/help/148-Getting-the-form-iFrame-code 

On regards of payment forms, you can only use a payment gateway in your form: 

https://www.jotform.com/blog/246-Create-JotForm-Payment-Forms-Using-Square 

https://www.jotform.com/help/221-How-to-Test-Authorize-net-Payment-Tool-on-Sandbox-Mode 

Let us know if you need more help.