If I use Flash when they cast Hidden Blade in combat, do they have to kill their own unit instead?

Ruling: No. Once targets are chosen when finalizing a spell, they cannot be changed. When Flash moves the targeted unit to another zone, it becomes an illegal target for Hidden Blade, so the spell cannot execute any of its instructions (can't kill the unit and its controller can't draw 2).

Nuances:

• This differs from Falling Star and Icathian Rain because those spells create reflexive triggers that target when they resolve, giving a priority window to respond before targeting happens
• Only spells/abilities that say "do this:" or "do this X times:" create a window where you can react before targeting happens
• Hidden Blade's "its controller draws 2" requires checking information about the target, which cannot be done if the target is illegal
• If the unit is saved by a replacement effect (like Zhonya's Hourglass or Sett's legend ability), the controller still draws 2 because the target was legal when Hidden Blade resolved
• For information to be referenced, the unit needs to either be on the board or have been moved to another zone by the effect that references its information