Square's physical gift cards are directly supported on KioskBuddy kiosks, the customer simply needs to swipe the gift card on whatever payment device you're using to power the kiosk. In this case, this implies either the Square Stand or an audio jack magstripe reader since swiping will be necessary for the gift cards.
The customer will need to select a Credit Card payment to be able to swipe a gift card.
For e-gift cards usage, there's a setting to enable gift card entry in the KioskBuddy app's payment settings section, as seen below in red.
You'll also want to enable the setting for split payments, which will allow your customers to use multiple payment methods per transaction, in case the gift card has an insufficient balance, also seen below in red.
You may need to update your KioskBuddy + Square Account permissions for the gift card entry setting to work properly. You can do so by navigating to the Account tab of the KioskBuddy Dashboard and pushing the Update My Square Account Permissions button.
When the gift card entry option is enabled, customers will have a new payment option that can be selected when checking out:
Selecting this payment option will prompt the customer to type in their gift card code: