Automatic switching of a constant current circuit
I would probably use an N type mosfet on the low side of your LED string and switch the earth end. Being on the low side will mean that you can drive it on/off directly from an Arduino output.

edit. Yes your diagram is fine, except... be sure to connect the Arduino's ground to the N-Mosfet's S terminal. 

edit2. That mosfet's Vgs is a bit high, so you may find the Arduino struggles to turn the mosfet on hard enough to avoid heating. What other n-mosfets do you have?