Scientific consensus seems to suggest that the nutritional quality of an egg is not solely determined by whether hens are kept in cages. Instead, it is influenced by factors related to being kept in a cage such as the hens' diet, overall health, and access to the outdoors. These factors collectively determine whether an egg is more nutritious for humans.

Environment: When hens are no longer kept in cages, they are able to spread their wings, move up and down multitiered aviaries, and perch up high like they would do in the wild to avoid predators. These more natural behaviors and more space for hens have been linked to healthier egg outcomes.
Diet: When hens are given access to the outdoors, they may be able to supplement their diets by foraging for insects and plants and may produce eggs with darker yolks due to higher lutein content and potentially more omega-3 fatty acids.
Organic: Hens fed organic feed and raised to certified welfare standards which don’t include cages are generally not exposed to synthetic pesticides or fertilizers and can produce healthier outcomes and more nutritious eggs.

One recent study from Poland that investigated the effects of different production systems on egg nutrient composition and found free-range systems may improve the yolk’s fatty acid profile and have a more positive impact on human health.