Zello will work with wired headset while in the background with the screen off. Just make sure to adjust these settings:
1. In Zello Options > Push-to-talk button> tap drop down menu and then press the wired headset button.
2. In Zello Options > Push-to-talk button action to Toggle
3. In Zello Options > Push-to-talk button check Handle headset buttons when in the background
Then you should be able to use the button on the wired headset in a toggle mode -- press once to start talking and agan to stop. Hold to talk mode will typically not work because of design of wired headsets-- they short the mic when button is depressed.
The same may work with Bluetooth headset however the compatibility vary between different Android devices and headsets.