Are shih tzu's prone to eye ulcers?

Are shitzu's prone to eye problems ie. ulcers

Yes!

As in any short-faced, large-eyed breed, the Shih Tzu has eyes that can easily be injured. Check them daily, and any time you see the dog squinting or rubbing at his eye. If your dog's eye is bloodshot, cloudy, or partially closed or has what appears to be a white dot in the pupil, take the animal to the veterinarian at once. Eye injuries can be very serious if they do not receive prompt attention; the sooner treatment begins, the more likely healing will be rapid and uneventful. A neglected eye ulcer can require surgery and even removal of the eye.

Afraid so! However you can bath them in a luke warm water with a couple of drops witch hazel. this is not a cure but it does help with the healing and reduces redness.