It's not related to calcium, from my understanding.
A brown egg is genetically a white egg (when you crack it, the inside of the shell is white) with a "spray" of brown coating on the outside. This spray accounts for the different shades of brown that different hens lay. Some hens apply a nice, even coating of spray. Some have a sprayer that gets, well, clogged for lack of a better term. Think of a can of spray paint when it's not shaken well. It spatters. This is pretty much what happens with some hens. Now, I'm not advising you to shake your hen
I found this one today from one of my Barred Rocks, I believe. I thought it was dirty, but now I just think it's pretty! She has just begun laying in the last few days. I love all the helpful info here.