IMHO King John was a bad king because he was a king who was a bad man. King John showed that he was a bad man by killing a bunch of kids.
In addition to killing his nephew Arthur, John is said to have hanged 28 boys aged 12 to 14, hostages from Gwynedd, at Nottingham Castle in 1212, and also about 1212 it said that one of John's officials hanged a 6 year old Welsh hostage.
Killing even one person who is a minor is enough to make someone evil in my eyes.