Can I configure voice dialer to be usable in hands-free mode? I use Cyanogenmod 11. It comes with an app called Voice Dialer. 

The only problem with it is that you have to click on it to use it, kind of defeating the whole point of it.

Is there a way to have it start when I press the "control" (I don't know how to call that button, it's used to pause music or hang up a call) on my earpiece?
Yeah bro there is an app on Google Play Store that can control your headset button actions, The app will help you customize the options when a button is pressed on your device, so with that app you can go to "all apps" and select the name of the voice dialer program, so next time you click on the heaset button, the mobile does work for u

link for app
https://play.google.com/store/apps/details?id=tvk.headvol 

if u find it helpful u can accept my answer