This is a misunderstanding which most of the people would meet. In fact most of the ultraviolet from sunshine would penetrate from the thin clouds. If the clouds are thick, it only would provide some protect in certain extents. The over isolation will lead to impaired skin even cornea. Most of the skin cancers is induced by the over strong ultraviolet. The high altitude places will have intensive sun burn, so even if you will feel cool on the top of mountain, your skin will get burnt sooner than the low altitude places. The ultraviolet and infrared in the sunlight is the chief culprit of your sunburned eyes. The sunscreen cream and sunglasses which coated with film will be the good choice for you to hide your eyes from sunlight.
Yes, a cloudy day your eyes can be sunburned. Dry eyes are caused by excessive UV-A and UV-B rays. Not only the sun contributes to the sunburned eyes, but also the UV radiation which reflects off snow and water can cause sunburned eyes as well. felt a few hours later. So if you feel your eyes dryness, itchiness, burning, tearing and heightened sensitivity after sunburn, you should go to see a doctor because high visible or blue light can contribute to a marcular degeneration and lead to blindness at worst. So in order to protect your eyes from harmful rays, you should wear sunglasses whenever outdorrs and look for the words 100% UVA and UVB protection and HEV,near-UV,or blue-light filtering.
Yes, it is possible for you to get sunburned eyes on a cloudy day because of the coming uv rays. The radiation of the uv rays will cause the damage of your eyes. You'd better wear the sunglasses when you go out, especially in summer. If you get the sunburned eyes, you could use the warm compress as the home remedies to protect your eyes to make them feel comfortable.