The definition of nutrient density you offer is similar to mine, but then the inclusion of protein as a nutrient makes no sense. If you include macronutrients in this definition, the whole term becomes meaningless -- in other words, protein has no place in a nutrient density equation. Eggs are very low on the nutrient density scale. To see how they rate based on comprehensive calculations, please review this:
https://www.drfuhrman.com/library/article17.aspx
For the record, on this scale that rates foods,the highest food tested received a score of 1000, eggs scored a 27 and white bread and white pasta scored a 21. Again, eggs are far from being a nutrient dense food.
For purposes of those who have diabetes, hypothyroidism etc., the key is not to take isolated beta carotene (which is an unhealthful practice), but to up serum carotenoid levels from fruit and vegetable intake. This provides mixed carotenoids which are much more protective and will be the best bet for optimum health.
I believe the best diet includes moderate to reasonably high levels of fat -- they just need to be from nutrient dense sources -- not refined products like oils. The sources for fats should be whole nuts, seeds and avocados. Those who have weight to lose probably should use a moderately low fat diet since they are burning their own body fat (so in essence, are on a high fat diet anyway for purposes of nutrient absorption).
As far as free range chicken living happy lives, that is almost without exception, a complete falsehood. Indeed, the term "free range" is not regulated at all when it refers to eggs -- so it can be used without actually meaning anything. Please review the following to learn more about what "free range" actually means:
www.eco-labels.org | Trivia Question On free-range chickens and eggs | By Umbra Fisk | Grist | Ask Umbra | 22 Mar 2006
Best,
Joey