The stigma concerning cholesterol has been fairly overblown or the data misinterpreted over the last few decades. Cholesterol is an essential and vital part of any diet because cholesterol is essential for the body (without it we are dead); what is important is the kinds of cholesterol one consumes, the source of the cholesterol and the overall diet. A breakfast of omelet filled with cheese alongside heaps of bacon, sausages and beans plus a "healthy" shower of salt will not do your health any favours, but not only because of the egg yolks. A boiled egg over whole-wheat bread is another matter entirely.
Furthermore, the overall health should also be considered. Studies have shown eating too many eggs has been associated with heart problems in type 2 diabetics and older males predisposed to cardiovascular problems (either due to obesity/overweight or familial history).
Abstract from an academic review done by the University of Manitoba:
Originally Posted by International Journal of Clinical Practice
For many years, both the medical community and the general public have incorrectly associated eggs with high serum cholesterol and being deleterious to health, even though cholesterol is an essential component of cells and organisms. It is now acknowledged that the original studies purporting to show a linear relation between cholesterol intake and coronary heart disease (CHD) may have contained fundamental study design flaws, including conflated cholesterol and saturated fat consumption rates and inaccurately assessed actual dietary intake of fats by study subjects. Newer and more accurate trials, such as that conducted by Frank B. Hu of the Harvard School of Public Health (1999), have shown that consumption of up to seven eggs per week is harmonious with a healthful diet, except in male patients with diabetes for whom an association in higher egg intake and CHD was shown. The degree to which serum cholesterol is increased by dietary cholesterol depends upon whether the individual's cholesterol synthesis is stimulated or down-regulated by such increased intake, and the extent to which each of these phenomena occurs varies from person to person. Several recent studies have shed additional light on the specific interplay between dietary cholesterol and cardiovascular health risk. It is evident that the dynamics of cholesterol homeostasis, and of development of CHD, are extremely complex and multifactorial. In summary, the earlier purported adverse relationship between dietary cholesterol and heart disease risk was likely largely over-exaggerated.
From the abbreviated abstract of another academic review from the University of Tennessee:
Originally Posted by Journal of the American College of Nutrition
... Several studies have examined egg intake and its relationship with coronary outcomes. All but one failed to consider the role of other potentially confounding dietary factors. When dietary confounders were considered, no association was seen between egg consumption at levels up to 1 + egg per day and the risk of coronary heart disease in non-diabetic men and women.