Can you exile a creature with a flicker effect in response to a destroy effect to save it?

Yes, you can use a flicker effect (exile then return) in response to a destroy spell or ability to save your creature. Magic uses a last-in, first-out stack system (CR 116.4), so if you respond to the destroy effect by casting a flicker spell, your flicker resolves first.

When the creature is exiled, it ceases to exist on the battlefield entirely. The destroy effect on the stack now has no legal target — or more precisely, the object it targeted no longer exists as the same game object. When the creature returns from exile, it is a brand-new object with no memory of the previous targeting (CR 400.7). The destroy effect then resolves but does nothing because its target is gone or invalid.

If the destroy effect says 'destroy target creature' it requires a legal, existing target on resolution. Since the original object no longer exists, the spell or ability is countered for having no legal targets (CR 608.2b). The creature safely re-enters the battlefield.

Example: Your opponent casts Doom Blade targeting your Grizzly Bears. In response, you cast Ephemerate on Grizzly Bears. Ephemerate resolves, exiling Bears. Doom Blade then resolves but its target no longer exists, so it's countered. Grizzly Bears returns to the battlefield at the beginning of the next end step, completely safe.

Note: This works against targeted destroy effects. Against non-targeted sweepers like Wrath of God (which destroy 'all creatures'), the flicker still works — your creature is simply not on the battlefield when Wrath resolves, so it isn't affected (CR 608.2c).