Why does my dog have a negative reaction to chicken based dog food/treats but not ferret food?

It's most likely not the chicken. The way to truly test the chicken theory is to feed the dog a bit of boiled unseasoned, defatted chicken and see if you get the same response as you do with the food. I've learned from experience that more dogs have issues with the plant components of dog foods. I have a dog that doesn't tolerate peas or legumes. I didn't realize how many of low grain foods contained peas and alfalfa as protein sources. It may take you awhile for you to find the offending source of the allergy.