EDIT: Turns out I didn't quite catch ALL the whines, though this mod, in its current form, removes most of it. There's still a slight, quick whine when idling. I have to track it down, and will update this mod and this edit when it's done

Thanks to LiquidBronze for walking me through this, and supplying a blank audio file in his "No More cash Register Sound" mod

This file replaces Dogmeat's constant whining noises to a blank audio file, thus making no sound. The voice (bark?) dog-actor must have been really convincing, because it triggers my dogs like crazy. Got really annoyed with it, but I liked having the dog. 

Step 1 

Go to: \Documents\My Games\Fallout4 open Fallout4.ini

Find sResourceDataDirsFinal=STRINGS\ edit it to  sResourceDataDirsFinal=STRINGS\, sound\

Step 2

Drop the file into \Steam\steamapps\common\Fallout 4\Data

Play!