We now have a few figs that turn into giants mid-dial like the High Evolutionary, Box, and Emperor J.

So I was wondering about special powers and how they are always on unless turned off by the player.

My question is this:

If the fig takes damage and becomes a because of his SP, am I allowed to cancel it even though it's not my turn and say the fig doesn't change?
Or since it's on my opponant's turn, is the fig "stuck" as a giant until my turn?

At the beginning of any action, you can cancel any optional power.
It stays off til the end of that turn.

So, when your opponent targets Giant Box over another figure's head, you can cancel the power, and LOF will become blocked. However, he will remain standard for the rest of the turn, meaning he can be charged by standard grounded pieces and he only ties up figs in adjacent squares.

This does not seem fair. Under your theory...

Giant is giant on my turn...I hit for damage.

On your turn, Giant hides behind standard characters so he cant be targeted.

On my turn, I am Giant again and can attack.

Are you sure it works that way? I would hope you can only declare Giant on/off on YOUR turn. And it stays that way for the WHOLE turn, so you can only change it on YOUR NEXT turn....

From the LoSH rule book (page 13):

If your character’s power is optional, it is assumed to be in effect during an action unless you cancel it at the beginning of that action.All special powers are optional. When a power is canceled, it is canceled until the end of that turn and resumes effect at the beginning of the next player’s turn.

So, yes, it does work that way.

When you give your character an action to attack Emperor Joker, that is an action. At the beginning of that action, Emperor Joker can choose to cancel Embiggen and now he is a regular sized character for the rest of the turn. At the start of the next player's turn (Joker's controller in a two player game), Embiggen turns back on and Joker is a Giant again... until he cancels the power, the power is Outwitted, or he loses the power via dial clicking.

Powers can be deactivated during any action, including your opponent's. The described scenario is perfectly legal.