Thanks for that additional info Arlene, very helpful.
Your vet is correct about the Persian breed tending to have runny eyes. Their flattened faces and large eyes tends to result in tears spilling over on to their fur instead of running down the ducts into their nose. The flattened face also can result in narrowed air passages through the nose and back of the throat, causing a pet to snore or breathe noisily, especially when relaxed or sleeping. Sometimes this can result in a squeaky noise, even when swallowing.
However, an airway "squeak" can also result from a "nasopharyngeal polyp", a benign smooth growth that can develop in the back of the throat, or grow down from the inner ear down in to the back of the throat. This is not necessarily a cancer (although cancers can also grow in those areas), they can sometimes be diagnosed by anesthetizing your kitty and closely examining the back of the throat, the back of the nasal passages and looking deep in to the ears for any abnormalities. The polyps can be removed by pulling them off at their base with a forceps, if they can be found. Sometimes they can be sneaky and very hard to find however.
Another cause of a squeaky noise associated with the respiratory tract can be feline asthma. In mild cases, a cat may not even look like they are having trouble breathing, but they tend to cough and some abnormal squeaks and wheezes can sometimes be heard with a stethoscope.
If he is able to eat and drink without it stimulating a lot of coughing and he otherwise looks perky and happy, then he can probably wait until your own veterinarian can see him. She may want to anesthetize him after she does her full physical exam, and examine the back of his throat, she may also suggest X-rays of the head, neck and chest. Best not to feed him the morning of his appointment so he will have an empty stomach.
If his coughing is steadily getting worse, or the squeak gets more high pitched, results in him making increased efforts to breathe, or he actually starts breathing with his mouth open........these are emergencies and you should consider asking your veterinarian to see him right away or take him to an emergency veterinary clinic.
I think the lion cut for Persians is always a great look! I hope this turns out to be nothing serious for him.