Google tells me that people search for an answer to the question: “Do British Shorthair cats have eye problems?” The answer is that they don’t. However, I think I know where this thought comes from. The British Shorthair does have a slightly brachycephalic head shape; slightly short and rounded. This has been developed by selective breeding since the beginning of the cat fancy in the UK as this is one of the first cat breeds of the cat fancy.
Persian cats are known to have eye problems by which I mean they have tear duct overflow and, incidentally, breathing problems. Persian cats have very round heads and a flat face. However, as mentioned, the British Shorthair does not have these problems.
For the sake of completeness, recognised medical problems for this cat breed as stated by Dr Ross D. Clark the VM in his book Medical, Genetic & Behavioural Aspects of Purebred Cats are (1) haemophilia B and (2) neonatal erythrolysis.
This is a hereditary deficiency in clotting factor IX. It is known as Christmas Disease. It has been reported in British Shorthair cats. Affected cats may bleed excessively after injury or surgery.
This is another disorder found more commonly in British Shorthair cats than in other breeds. It refers to the destruction of red blood cells in newborn kittens due to a difference in blood type between the queen (mother) and her kittens. Most cats have blood type A. This condition sometimes occurs if kittens are born with blood type A to a queen with blood type B.
Russian British Shorthairs
In my experience Russian British Shorthairs are bred to a more extreme confirmation than the American variety. This may cause some health issues but I have no knowledge of them. There is nothing on the internet or in the books that I have which supports the idea that this breed suffers from congenital defects other than the ones stated above which are probably quite rare.