Can I react with Zhonya after my unit dies so Zhonya doesn't activate?

No — not if Zhonya's is already in play. Zhonya's Hourglass is a mandatory replacement effect, not an optional trigger (FAQ #5836). It doesn't have a "you may." When a friendly unit would die, Zhonya's must kill itself instead and save the unit. You cannot let the unit die to save the Hourglass for later.

If Zhonya's is hidden (facedown at a battlefield) instead, it's a different story. A hidden card has no active replacement effect yet — you simply choose whether to reveal it or not. If you never reveal it, it stays facedown (and will be discarded if you lose control of that battlefield). But there's no way to "react after the unit dies" and have Zhonya's enter play without saving it — either you reveal it in time (before the death event) and it must save the unit, or the death already happened and Zhonya's enters too late to do anything (FAQ #9663).

So in short: if it's revealed, mandatory. If hidden, just don't flip it.