Yes, they actually do both. They contain many enzymes, but their name comes from the fact that they break stuff down through oxidation by producing hydrogen peroxide. They contain an enzyme called catalase (if I remember correctly) that breaks down the H2O2.
Yep, they oxidize and produce peroxide, which is toxic, which they then break down, making water and oxygen gas if I recall. I think we dumped some peroxide on a pigs liver to see the catalase reaction.