Yes and (possibly) no. If they only have one character in battle, then yes, they gain special initiative. If they have more than one character and you're not removing all of them, I believe they only get initiative if after the removal they would be losing the battle.
*Edit*
Definition of Special Initiative
When you are losing the battle by removal, you are granted special initiative to play an Enhancement that will interrupt or negate the card that is causing your character's removal. You are considered to be losing by removal when an opponent's special ability, or a game rule that has been triggered by an opponent's special ability, would leave you with no character in battle when the special ability has completed.
So if they have more than one character in battle and you are not removing all of them, they only get initiative to interrupt/negate you if they would then be losing by the numbers (or ignore, stalemate, etc.), but it would be normal initiative.
