It is absolutely true that Apple will replace the iPhone 6 screen for $109--and that this includes a new home button/touch id sensor and the reprogramming of the phone for the new touch ID.
This is your only option to regain touch function if you have damaged your original touch ID flex.
Read more about this coverage here, and no it doesn't require Applecare +
http://support.apple.com/kb/index?page=servicefaq&geo=United_States&product=iphone&select=WARRANTY__PRICING