Does Turn to Dust on an equipped unit still kill the gear that is equipped the following turn?

Yes. A gear hit with Turn to Dust while attached will be killed during your next Beginning Phase.

Here's the key distinction: when a gear is attached, its printed rules text becomes inactive (Rules 135.4, 718.2). However, Turn to Dust grants the [Temporary] keyword — and granted rules text remains active while attached. The granted [Temporary] triggers normally in your next Beginning Phase, killing the gear and detaching it from your unit. Any stat bonuses or effects the gear was providing cease immediately at that point.