Stellwart goathert can destroy bomb

Yes — Stalwart Goatherd's play trigger can kill a Bomb gear token, including an opponent's, because it says "kill a gear" with no friendly restriction.

Bombs are tokens, and tokens have no cost — but their cost is treated as 0 for all purposes (Rule 185.3.a.1, with 206), so a Bomb comfortably satisfies "Energy cost [4] or less." The Bomb lives at a battlefield and Goatherd arrives at your base, but the trigger has no location restriction, so that doesn't matter. The trigger is a "may," so you can decline.

One thing to watch: killing a Bomb fires its Deathknell — deal 2 to an enemy unit there and kill all other Bombs its controller has there. "There" is the Bomb's battlefield, so if you have a unit at that battlefield, that 2 damage lands and can kill it; and each other Bomb killed that way triggers its own Deathknell in turn.