I think the OP may have been asking (and what PixieChixie was saying) what the health benefits are to the chickens, rather than to people who eat grass fed animals (or, uh, grass).
Chickens are omnivores and as such a balanced poultry diet will include bugs, small animals (mice/snakes/etc), grasses, leaves and other plant material. Given the opportunity to free-range the birds will eat to satisfy any lack in their diet.
The benefits to eating the correct diet for a body type (such as 100% grass-fed beef as cows evolved to eat grasses pretty much exclusively) include highly functioning immune systems leading to low incidence of illness, healthy growth of hair/feathers/fur, effective digestion and successful reproduction. Any animal with a balanced diet will have an immune system that keeps parasite load at a manageable level.