Why is my barn cat's eye swollen, red, and constantly tearing?

I am sorry to hear about Finn's eye. It is very difficult to truly evaluate an eye based on a photo. All I can really tell from these pictures is she seems to be squinting the eye with increased tears and there may be some minor swelling/inflammation around the lids. We can have a variety of possible causes for eye problems like this. She could have scratched her eye and this is causing irritation, she could have gotten some debris into the eye that is causing irritation, we can have eye infections (like people getting pink eye) as well in pets. These would be the most common causes of eye problems in a cat such as yours. I would recommend flushing her eye out with a sterile eye wash solution (you can purchase sterile eye wash at the pet store or drug store) in case she does have some debris in the eye. If the issue persists or is worsening then an exam by a vet is in order. They can better examine the eye with an ophthalmoscope and can put special stain on the eye to check for ulcers or scratches. Depending on their findings they may prescribe medicated drops or ointment. Best of luck to you and Finn.