How did reanimated Haku and Zabuza die?
Neither Haku or Zabuza were "killed" in the sense you describe. In the same episode you referenced, they were both temporarily immobilized: 


Haku was immobilized by shielding himself in front of Zabuza when attacking was attempting to strike him with Chidori
Zabuza was immobilized after Kakashi successfully hit him with a Chodori.


As they were immobilized, this gave Ensui Nara an opportunity to hold them with the Shadow Bind technique


  After Zabuza cuts through Haku, Ensui told Santa that Kakashi only caught one enemy and immobilised Haku as he regenerated. When Kakashi successfully connected his shadow to Zabuza's, he immediately immobilised him as well.


In this Youtube video, you can see the shadow line holding two shinobi down. This gave another shinobi to use the Cloth Binding Technique to capture them.