From a quick read? No.
The Unchained Barbarian's Rage reads as: "While in a rage, a barbarian gains a +2 bonus on melee attack rolls, melee damage rolls, thrown weapon damage rolls, and Will saving throws. In addition, she takes a –2 penalty to Armor Class"
The Primal Hunter alters rage to grant a scaling bonus on attack rolls with ranged weapons while removing the Will save bonus and allowing Stealth checks.
Therefore there's no interaction between the Bonus to hit/damage from an Unchained Barb's rage and the to hit bonus from a Primal Hunters rage, nor is there any reading to indicate that the Primal Hunter functions with the Unchained Barbarian in regards to damage, as that was dictated by the Barb's STR Bonus.
They wouldn't put a note in for it, since Unchained doesn't utilise or is necessarily compatible with player companion material.
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I'm having a hard time seeing how the damage bonus wouldn't apply if the ranged weapon in question is a thrown weapon. The +2 attack bonus with ranged weapons the primal hunter gets is in addition to everything else a barbarian gets from raging. So a 1st-level unchained barbarian with the primal hunter archetype should be getting a +2 to attack and damage rolls with a thrown weapon, but only a +2 to attack rolls with all other ranged weapons. It's no coincidence that the bonus increases at the levels where a barbarian gets greater rage and mighty rage.
I'm glad it's been FAQ'd, but I don't think it should be too difficult to adjudicate.
EDIT: Pathfinder Society says that the unchained barbarian qualifies for any archetype that doesn't alter rage and uses the urban barbarian as an example, so that would exclude the primal hunter as well. So it's not even possible to use for Society play. However, for those outside of that ruleset, it would need to be addressed, so I FAQ'd it as well.