If my iPhone is running iOS 10 and my home button breaks, do I lose the ability to make an emergency call?
You will need to turn on AssistiveTouch prior to needing to call Emergency.  

Go to Settings -&gt; General -&gt; Accessibility -&gt; AssistiveTouch and turn it On.  This will show a screen overlay of several options, including the home button.  (You can also just tell Siri to turn on AssistiveTouch)

Whenever you need to call Emergency, wake the iPhone the usual way and then tap the AssistiveTouch screen overlay and swipe to bring-up the ability to call Emergency.