If I cast shuriken flip and my opponent responds with flash, do I still get to move my unit?

Yes — the move still happens. Flash can only move the caster's own friendly units, so your unit stays exactly where it was and stays a legal target for "move a friendly unit."

The two instructions on Shuriken Flip aren't linked (no "if you do" between them), so the mandatory move executes independently of the damage half (Rule 359.3.e.14.a). What Flash can blank is the "Deal 2": if your opponent moves the enemy unit you targeted off the battlefield, that instruction mistargets and no damage is dealt (Rule 359.3.e.5) — same as the Incinerate/Flash ruling (FAQ #2967). You still can't skip the move, though; it's a required target, not optional (FAQ #11883).