Iirc, Haku gave up because acknowledging Naruto's spirit and wanting to teach him the toughness of the Shinobi world. The moment he realized he could still help Zabuza, he parried Naruto.
And Naruto's damage is kind of inconsistent in part I. In the valley, he parried Sasuke's Chidori a bit, so iirc it missed Naruto's heart, and Naruto could regenerate. When Kabuto damaged his heart during Sannin showdown, Naruto was about to die, and Kyuubi could only watch. Since Naruto won't start in 1st Tail form, I don't see what's stopping Haku from neck-incapitating him the same way he did Zabuza.
But I guess depends on the mindset how quickly Naruto will access Kyuubi Chakra.