"Open loop" gift cards should trigger Chargeback Fighter to snap an image. These run through VISA and MasterCard payment systems.
Other gift card solutions may or may not work. There are many different gift card solutions (Magstripe, chip card, QR code and many other gift card applications) that work on the Clover platform. There is no clear 100% yes or no answer without testing them. Since there are so many different types of gift card options, you will need to test gift cards that you accept through the Clover device to verify the answer for your specific situation.