Yes food makes a huge difference. Corn will increase yellow intensity but also access to fresh vegetation like grass and a varied diet is likely best. Another thing I think is a big factor is the frequency of laying. This interacts with diet too. Hybrid layers produce more eggs then heritage breeds but often these eggs are less nutrient dense on average. If you think about it, it just makes plain sense. You can't have a chicken that produces huge amounts of meat and quality eggs at the same time... 

 Agree, hybrid hens are working overtime to pump those eggs out and they don’t have the same nutritional profile as the eggs from hens that’s been fed well and cared for