If I have hidden a Flight or Fight and my opponent casts Hidden Blade, can I react to move my unit back to base and save it? Or does the Hidden Blade effect still happen even though my unit would not be at a battlefield?

Yes, you can save your unit this way. If you move your unit to base in response to Hidden Blade being played (before it resolves), your unit changes zones from a battlefield to base. A target is illegal as the spell resolves if it no longer meets the targeting requirements of the spell, or if it has changed Zones to or from a Non-Board Zone. Since your unit is no longer 'at a battlefield' when Hidden Blade tries to resolve, it becomes an illegal target. If any of the spell's targets are no longer legal, those targets are unaffected by the spell as it resolves. The result is that your unit is not killed (it's safely at base) and your opponent does not draw 2 cards (because the target became illegal, the entire effect fails to apply). This is a valid and effective defensive play.