Brown eggs are
more nutritious than white eggs
False
The color of the egg is determined by the breed of the hen that laid the egg. (The color of the shell she uses to coat the egg) The color of the egg (be it blue, green, blue, brown, or spotted...) does not change the nutritional value of the egg inside.
That said, the color and flavor of the yoke can be affected by what the hens eat. A free range egg will typically be far more orange than a regular hen's egg.