The expression “hair and hide” means: The whole works; every part; the entirety.
The first part, hair and hide, has long been used in the same sense. In fact, five hundred years ago, in the metrical Life of St. Cuthbert, we find: “pai were destroyed, bath hare and hyde.”
The second part, horns and tallow, is, we suspect, a fairly recent American additive probably of the wild-woolly-West school of literature to impress youthful readers.
When the earlier part is reversed and used negatively, as in “I have seen neither hide nor hair of the cat since yesterday,” the sense is that the speaker hasn’t seen any part of the animal, and this usage dates back apparently little more than a century.