Does the diet and raising standard of hen have more impact on the egg nutrition than the breed or shell color?

Yes. Breed and shell color (brown/white) have little to no impact on the quality and nutrition of the egg compared to the diet of the hen and how it was raised. Any hen, irrespective of the breed, can still be given processed synthetic feed and be exposed to chemicals and antibiotics. What matters more is how the hen was raised and what it was fed.

Check out various references that show how diet and raising standard affect the nutrition of the egg:

Finding	Reference	Link
Shell colour is determined by breed/genetics and is not an indication of nutritional value.	USDA / MSU Extension	1
Impact of egg colour, hen strain, and molting on nutritional composition was limited.	Anderson, 2013, Poultry Science	2
Hen diet can significantly affect egg nutrients such as fatty acids, fat-soluble vitamins, vitamin B12, iodine, and selenium.	Naber, 1979, Poultry	3
Compared with caged hens, pastured hens’ eggs had 2× vitamin E, 2× long-chain omega-3 fats, 2.5× total omega-3 fatty acids, and less than half the omega-6:omega-3 ratio. Vitamin A concentration was also 38% higher.	Karsten et al., Renewable Agriculture and Food Systems, 2010	4
Hens exposed to sunlight produced yolks with 3–4× higher vitamin D3 than indoor hens	Kühn et al., 2014 — Nutrition	5

Links:

1	https://www.ams.usda.gov/publications/qa-shell-eggs
2	https://pubmed.ncbi.nlm.nih.gov/24235237/
3	https://www.sciencedirect.com/science/article/pii/S0032579119347078
4	https://www.cambridge.org/core/journals/renewable-agriculture-and-food-systems/article/abs/vitamins-a-e-and-fatty-acid-composition-of-the-eggs-of-caged-hens-and-pastured-hens/552BA04E5A9E3CD7E49E405B339ECA32
5	https://www.sciencedirect.com/science/article/abs/pii/S0899900713004474