Can you save a unit from dying using Zhonya's Hourglass from a different battlefield than the one the unit is dying on?

Yes. Zhonya's Hourglass can save a unit on a different battlefield than where it was hidden — the battlefields don't need to match.

Here's why: when played from Hidden, gear is still played to base (not to a battlefield), and Zhonya's replacement effect ("If a friendly unit would die...") doesn't target a unit. Because there's no targeting, the Hidden restriction that limits you to "this battlefield" (Rule 811.1.d.2) never applies. The effect simply looks for any friendly unit about to die, wherever it is.