It turns out some of the bog bodies do/did have red hair, but by no means is it a majority of them. The issue also seems to be complicated by the fact that the conditions in the bogs can remove dark pigments in hair making it appear more red than it would've originally been.
The two most noted red-haired bog bodies I've came across so far are Yde Girl and Neu Versen Man. Neu Versen Man is also known as Roter Franz meaning Red Franz in English - so called because of his red hair. Yde Girl is the bog body of a teenage girl who had reddish blonde hair. Interestingly, one side of her hair was thought to have been shaved off before she died - although it's been suggested that this lack of hair may be due to the way the body decayed, with one side of the head being more exposed to oxygen than the other.
I also came across this interesting article while I was searching. It concerns mummies rather than bog bodies and tells of medieval bodies found clad in copper masks in Siberia.
"The best preserved mummy was a red-haired man found in a wooden sarcophagus. He was covered chest to foot in copper plate and was laid to rest with an iron hatchet, furs and a bronze head buckle depicting a bear."http://news.discovery.com/history/archaeology/siberian-mummies-in-copper-masks-post-mystery-140416.htm