HI cats73w -
This is usually just a general sign of not feeling well or else conditions that are localized to the eyes. So, yes, he could have intestinal problems and this could be a symptom of this, but it could also indicate a lot of other things.
If the white parts of the eyes look red or inflamed, or there is any discharge from the eyes, or if your kitty wants to hold his eyes closed,especially when in bright light, these would suggest problems specific to the eye.
If there are other symptoms that you are seeing (vomiting, diarrhea, decreased appetite, coughing, sneezing, changes in water consumption) then these may be clues and these may need to be persued further
It is good that your kitty tested negative for Feline leukemia and immunodeficiency virus as these are both very serious diseases, so that was a good place to start. However, additional testing will likely need to be done to try to find out what exactly is going on. What these tests will be will depend on what other symptoms are present and what is found on the physical examination.
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If the eyes are red and irritated, you should have him seen by a vet as soon as possible as eye problems can worsen quickly.
If he is showing some other symptoms, then treatment will depend on what those symptoms are.
If he is acting normal in every other respect (eating, no vomiting or diarrhea, no coughing or sneezing or any other symptoms) you can watch to see what direction this goes.
I would not give antibiotics since we don't know if this is an infection and over use of antibiotics will make it more likely that a pet will develop a resistant infection.