Most of the fat in a hard boiled egg is contained in the yolk. The yolk provides about 55 calories worth of combined fat and protein. Egg whites, on the other hand, are packed with fat-free nutrition. You'll consume 4-5 grams of protein, just 17 calories and virtually no fat in a single large egg white.
Vigorous coughing is one possible reason for burst eye capillaries. When a capillary in the white part of the eye bursts -- called a subconjunctival hemorrhage -- the appearance of blood can be rather dramatic and alarming. Often, there is no history of trauma to the eye -- the blood simply appears spontaneously.
There are several causes of cloudy eyes (opacity, catarracts). It can be caused by any number of systemic illnesses, such as the various organisms that also cause diarrhea, diabetes, etc. Also, deficiencies such as Vitamin A can lead to cloudiness. The most common cause, however, is trauma of some sort.