There is a major difference between brown and white eggs: The color. It all depends on the breed of the chicken from which the egg came from. It really depends on what you feed the chicken and the condition of the hen that controls the taste and quality of the egg. The color makes no difference and don't believe anything positive or negative based on the color.
The old adage that brown eggs come from brown birds is not quite true. You actually have to look at the feathers by their ear holes to know if they will give white or brown eggs.
We used to have some Plymouth Rocks and Leghorns when I was a kid and the Rocks were used for stewing hens and the Leghorns for fryers and layers. The Rocks were brown but they were larger and ate more. The eggs were a bit larger but tasted the same. I suspect that commercial egg producers found that it cost more to feed brown egg chickens so they used the smaller variety and charged more for brown eggs.
My mother grew up in England and while she didn't see an egg for almost six years, during rationing, she almost always saw brown eggs until she came to the U.S.