Voltage from Neutral to Ground
With an open neutral at some point it would not be unusual to see a phantom voltage close to line voltage. The reason to have a black and red hot to an outlet is to have a switched outlet (when there are only 3 + ground) the jumper on the receptacle/ outlet needs to be broken for it to work but this is normal. 

I would check and see if there is a bit of slack at the location you thought the white could be broken because I believe your neutral is open and the hot wires in close proximity to to your open neutral is creating a phantom voltage to ground.