Do I need to peel a red kuri squash?
Most Japanese varieties of pumpkin do not need to be peeled. More precisely, most Japanese are content to eat most Japanese varieties of squash unpeeled (maybe rough peeled where knobs are present). I can't really recall treating red kuri any differently, but I don't find it very often, so I can only speak from limited experience.

The typical preparation of any squash in Japan is nimono, in which squash pieces are gently simmered with kelp, soy sauce, mirin, sugar and salt until tender. The regular kuri variety (the green one with some occasional knobs) is well suited to this, but is even accommodating of being deep fried or roasted skin-on, and I've used that variety for soup and other preparations with the skin on. I would be surprised if red kuri weren't at least suitable for nimono skin-on, as I don't think the Japanese market would be particularly accepting of a squash that you would have to skin.

There is, for what it's worth, a site that suggests that the skins are harder on red kuri than normal kabocha varieties, and that the skins are not eaten. And the same site says that regular Japanese kabocha skin is not eaten, so they appear to disagree with basically all of Japan.

At least one soup recipe in Japanese for the red kuri squash says that the skin should be left on. 

Anyway, the short answer would be: Try it and see. The worst possible outcome is a texture that you're not fond of, or perhaps some bitterness.