Well, my iPhone 6s was having this problem too. Random shut downs with battery anywhere between 20-50%. Could not restart phone without plugging it in, at which point there would magically still be tones of battery power left. Had the problem with iOS 9.3. Upgrading to iOS 10 did not help.
So anyway, I submitted to have the phone mailed off for repair. I packed the phone up in the box they sent me and followed all instructions. Only after mailing the box back did I think about the fact that the box seemed a little large for my iPhone. it is almost like they sent me a 6s Plus sized box for my 6s. Has anyone else mailed off their phone for battery replacement? Did the box they mail you fit your 6s perfectly or was it a bit oversized like for a 6s Plus?
Anyway, after 9 days I finally got a phone back.....accept it isn't my phone it is a refurbished phone. Apple's note said they had to replace the screen, logic board, and battery so they just sent me a "new" refurbished phone back instead. Glad to finally have my phone but kind of bummed that I got a refurbished replacement instead of my original phone back.
The phone I sent to them was in immaculate condition and shouldn't have needed a screen replacement. I wonder if they sent me the wrong sized box to pack my phone in and the screen got damaged during shipping to them or something....
Would love to hear back from anyone else who had their 6s repaired by mail!