Nope. All 360 peripherals have a Security IC in them and without this IC nothing will work on the 360, except keyboards or whatever M$ allows, even 3rd party controllers have these ICs in them.
The only way you'd get that Headset to work on the 360 is to buy the Wireless 360 Headset, gut them both and install the insides of the 360 one into the Logitech shell, which may or may not be possible, depends on how much room is in the Logitech one.