In theory, there should be no difference between brown & white eggs, but when I visit the farms where the brown eggs are produced, at least here in Michigan, the chickens are out in the yard, scratching in the dirt and eating kitchen scraps like carrot peelings etc. So from my experience brown eggs are probably more nutritious.
It is unfortunate indeed that terms like 'free range' mean nothing, because when I give presentations on farm animal cruelty people start off making fun and by the time they're done watching the videos they don't even want to think about it the next time they eat a pork chop or steak or egg. There is nothing funny about a hen wired to her nest, never getting up from it for her entire life; with her upper beak painfully torn off and birth so the "farmers" can shove seed in faster to produce more eggs. Male chicks often "useless" depending on the farms goals, are ground up alive. The fact is we don't need any animal products to live and there is plenty of evidence we are healthier without them. But realizing that people will not give up meat or whatever, I just ask: can it least be produced as cruelty free as possible?