Official Microsoft Xbox 360 Wireless Headsets (Bluetooth & 2.4 GHz) do not have a dedicated hardware mute button on the earcup.
Many licensed (Turtle Beach, Astro, PDP, Razer) and unlicensed 360 headsets do include a flip-to-mute boom mic or inline mute switch. Check your specific model.
The official wired headset that plugs into the controller has a mute switch on the inline volume control module.
Tip: If you mute often, a third-party headset with flip-up mute is far more convenient than navigating the Guide menu.