If I have Zhonya's Hourglass and equipment on my unit and they kill the unit triggering the hourglass, does my equipment fall off the unit?

No, your equipment does not fall off.

According to RiftJudge FAQ #5370 and FAQ #2765, when a unit is saved by Zhonya's Hourglass, the unit never actually dies and never leaves the field (it is recalled to base, which is a board zone). Because the unit remains in play, all temporary effects, including attached equipment, remain attached to it.

As stated in Rule 109: "Whenever a Game Object changes zones to or from a Non-Board Zone, all Temporary Modifications of all kinds cease to be tracked on it in all capacities." Since Zhonya's Hourglass recalls the unit to your base (a board zone), the unit does not change to a non-board zone, and your equipment stays attached.