I went back to my first Mudbox book since I can paint now. Here's my modeled egg and the reference image I want to emulate. I chose the right shade of brown and easily painted directly onto the egg.
After the base coat, I chose a lighter shade and used a dotted stamp to break up the solid color. Then I chose a darker red for the larger dots. I used the same dot stamp but increased the size. These three steps were done on separate layers in case I needed to go back and tweak them individually.
An alternative approach is to use the actual photo of the egg as a stencil. You can't see it here but the photo stencil appears overlaid on top of the egg and disappears when I start painting. Whatever is in front of my brush gets painted onto the egg. The stencil is static so I have to keep rotating the egg to apply the paint which can lead to a repeated pattern forming. With that done, I took all the painted layers and adjusted their opacities until I was happy with the result. Along with painting colors, I can also paint shininess on the egg. If you zoom in on the egg on the right, you can see the shininess I painted on using the dotted stamp again. Below I have the final result. Mudbox has some filters you can apply so I added in a depth of focus look.