Eggshell color does not change nutrients in the egg. The color of the shell is simply decoration and is determined by which breed of chicken you own.
A little known fact: You may be able to tell the shell color by the hen’s earlobe. Hens with white earlobes typically lay white or lightly tinted eggs. Hens with red earlobes most commonly lay brown eggs.
Info obtained from countrysidenetwork.com