Divine, but that might be a mistake or a cut corner
Nothing in the ability says that you cast your patron spells any differently from your other ranger spells; it just says you add those spells to your spell list. Since your ranger spells are otherwise divine, these spells are too.
On the other hand, the ability describes itself as “A witchguard learn[ing] something of arcane magic from the witches he defends.” So it might be that this was a mistake, and the intent was that these spells count as arcane. However, since a ranger typically wears light armor, arcane spell failure would be a problem if those spells became arcane. They could waive ASF for those spells, à la bard, but then maybe that becomes a lot of text to describe this ability, when it really doesn’t matter very much. The differences between arcane and divine spells are very few, and the biggest one—arcane spell failure—is one you’d expect to have waived for a ranger. So you could spend a whole lot of page space explaining that the spells are arcane, but they don’t trigger arcane spell failure, and therefore... what? There’s no particular difference at that point, so why spend that space? So my guess is they just didn’t bother because it wasn’t worth the effort, but they “should” be arcane, in some sense.
If you are looking to qualify for something that requires arcane spells, I would just ask your GM if you can take it. Most of the time, there’s no real reason for such things to be limited to arcane spells in the first place, and I, for one, would rarely have a problem allowing it even for characters that are unambiguously divine. For these witchguard rangers, who know “something of arcane magic,” I’d be quite surprised if any GM objected.