Does Square support custom payment type

We have a mobile app that we are looking to add ordering feature.
Let say users pay with credit card via the square payment api, our kitchen staff can see the order on the Square tablet.

Beside square payment, we have a custom payment solution we are looking to add. Let’s use Paypal for example. If users select Paypal as a payment option, can the paid status be pass to Square? I am looking to manage orders paid via Paypal through square as well.

At this time, to be able to push orders to the Square POS or Dashboard, the orders have to be paid for via Square, not a third-party payment system like PayPal.