A quick regular google search produced, as expected, mostly garbage (material from advocacy groups, usually vegan or pro-"natural" stuff), but the one, not peer-reviewed
test I saw , did measure nutrient content in the eggs directly (for their own eggs, they just took the factory-raised eggs numbers from the USDA). No big surprise, they disagree with the American Egg Board's claims that nutrient content is not affected by free-range vs floor vs cage raised. I do believe the "MotherEarthNews" people misinterpret those claims, as I imagine, the amount of freedom of movement each chicken gets should not affect nutrient content in eggs significantly, but I do believe a difference in diet
could make a difference there; though the American Egg Board's claim does not touch this issue.
The one thing I am curious about is whether the egg laying regiment of factory-raised hens affects nutrient content in any significant way.