Bluetooth headset button replacement
It looks plausible.

Measure the button size, and try to find some at a distributer, look at their datasheet to ensure their is nothing weird happening (like a hidden order connection). That will also tell you the pins configuration.  

You might get more confident by practicing on similar buttons, so order some. . 

If they are big enough, tack-solder them onto some stripboard/veroboard and practice the modification on them, as well as driving the signal. At least hold them with blutak or hot glue. Practice on them to get a feel for what you are doing. Too much heat might damage the PCB, so try to enue you can do the job with a few seconds on heat.

Use a voltmeter, very carefully, to figure out what the switch is doing. It will either be 'high' (say 3.3V), and when pushed goes 'low' (say 0V), or vice versa.

If it is a simple signal pull-up or pull-down, then you could do that with either an N-Channle or P-Channel MOSFET, which are easy to drive from an Arduino pin, or an NPN/PNP transistor.