what happens if I apply a negative voltage across a zener?
It will go below ground one diode drop but the 10k resistor will limit the current into the zener and hence into the circuit that may be damaged so, the only question you have to ask is "how much current can my input take before it is damaged".

My gut feeling is that on most devices (MCUs op-amps and the like) 10k from +/- 6V is not going to damage anything but the most exotic of devices so, you are probably going to be OK.

Check your data sheet about how much current you can stuff up an input.