If I have only one unit, and my opponent bellows breath. If I flash to the base does my unit still die?

Yes — your unit still takes the damage and dies (if it has 1 Might).

Bellows Breath targets the units themselves, not their location. The "same location" is a targeting restriction checked when the spell is cast — and with a single target, that condition is always satisfied. Flash moving the unit to base doesn't undo the target; the spell tracks it to the base, which is a valid location, and deals the 1 damage on resolution. (If the unit has 2+ Might it survives with 1 damage marked.)