Death triggered abilities trigger when Hushbringer dies
Abilities that trigger upon a creature dying trigger when Hushbringer dies.
Steps to reproduce:
1. Wish for and play Hushbringer, Nightmare Shepherd.
2. Destroy Hushbringer.
Nightmare Shepherd's ability triggers.
Abilities do not trigger when Hushbringer dies.
Abilities also trigger for creatures that die at the same time as Hushbringer, such as Garrison Cat dying with Hushbringer.
According to the ruling for Hushbringer: "A creature entering the battlefield or dying at the same time as Hushbringer (including Hushbringer itself entering or dying) won’t cause abilities to trigger."
This issue does not occur when Hushbringer enters the battlefield.