Yes indeed! There are three factors in what color a chicken egg ends up being; the shell, itself (either white or blue), the outer coating (which can be anything on a spectrum from nothing (so pure white or blue) to super thick (so chocolate brown or olive green) and speckled or evenly coated, and the bloom (heavier bloom makes browns look pinkish or purplish and greens look grayish, and bloom can also cause white, pinkish, or purplish speckles). The genetics for shell color are simple, just one dominant gene, but the rest of it is, phew, a whole lot to try to decipher!
That's the one thing I can for sure say is a down side to feather-footed breeds. In the spring, when everything is muddy and the ground hasn't had the chance to dry out and stabilize, those foot feathers paint mud all over my eggs no matter how clean I try to keep the nests!
Diet likely plays a part, and a bird who is sick very often does have a messy bottom, but there must be more to it. All of my adult birds are on the same diet, and some will have clean bottoms and some, though perfectly healthy, will have dirty bottoms, even birds that are full-time contained so cannot be getting into something while free-ranging. I've never been able to figure it out, myself.

Just gotta take the scissors out every now and then and trim the mess out so that it doesn't cause an issue.