Hydrogen peroxide is a strong oxidizing agent in a 90% water solution. Rust in a bottle. It always gives me the heebie-jeebies to think of pouring that in a steel gun barrel, smooth or rifled. It will also bleach out the wood if it seeps into it, as StumBuff noted. There's plenty of other things to use that work as well with less danger; tap water & a bore brush or mop, for instance. Just 'cause it sudsies up doesn't mean it's helping cut the fouling. Warm water and Murphy's Oil Soap cut that just fine. Now, if you had a lot of bacteria in your barrel, it would be worth using it, maybe.
The ammonia in washer fluid/Windex will attack brass parts, causing stress cracking with enough exposure. Chlorox attacks stainless steel the same way.