We had a Black-Walnut tree that was next to our back terrace growing up. And have one now in our back yard. When I was little, my mother would make me gather the walnuts (in their green hulls) and get them off the terrace. What would happen otherwise is that they would get broken open and the inner covering of the nut itself -- the shell, I guess -- would stain the flagstone. And if you got some on your hands -- it was really hard to get out.
My friends and I loved to take hammers and open up the nuts to eat them! But, they sure are hard to get into. We also had one at the school I went to as a little girl. We loved finding the big green hulls, and opening up/eating the nuts but I remember my mom getting mad at me for coming home with stained clothes.
Maybe there is not an appreciable amount of Iodine in Black-Walnut s, but there is definitely something that gives them excellent staining properties!