Micro switch that closes circuit twice, on press and on release
If you can live with using a regular pushbutton in your headphone plug and have access to the interface behind the plug's socket, here's a little logic diagram that might do the trick:


                                                    +------+
                                       +-> Delay -> |      |
                                       |            |      |
Switch (w/ Pull Up/Down) -> Debounce ->+            | EXOR |-> Pulse on Edge
                                       |            |      |
                                       +----------> |      |
                                                    +------+


The pull-up/-down resistor and everything else would be behind the socket.

You use your switch with a pull-up or -down resistor and debounce the output. The clean signal is fed into an Exor gate - using a delayed and a direct line. Every time there is a positive or negative edge when you press or release the switch, the Exor will have two different input signals as long as the delay time, giving you a positive pulse.

The delay can probably be as simple as an RC combination, maybe you can even create a clever debounce circuit that outputs both the direct and the delayed edge.

This simple logic circuit can likely also be implemented using a programmable logic device or a uC.