I'm already seeing a number of products proudly claiming they don't use GMOs.
Edit: (sorry for changing my comment, I add the original below).
You are picking on the wording. There must be a succinct way to say non-genetically modified animal.
If it's GMO: GMO-Chicken. Seems succinct as well.
Non-GMO Chicken. Least succinct of the 3.
"Has excrement" would certainly be perceived as negative, but the effect on my shopping of "has no excrement" would be a lot more negative (trying to find this sticker on everything I purchase, for example).
If informed people won't do something, the answer shouldn't be to keep them in the dark, no matter how much you disagree with their reasons.