I would think on or off will not matter. . . . .but, yes, if the device itself actually gets that cold you could damage it. But, Realize that IT would have to BE below the stated temperature. . . .if it starts at 'room' temperature and is outside for a few minutes -- especially if it's in a cover or being carried in a bag, it's not likely to get that cold, even if that's what the air temperature is, unless it's in those conditions for an extended period of time. I am not a thermal engineer, but I wouldn't think 10 to 20 minutes would do it unless it was left completely exposed, alone and naked.

Still, if I was going somewhere and I
knew ahead of time it would be that cold and I'd not be able to keep my Kindle with me, I think I'd elect to leave the Kindle at home.
Now, if it does get that cold despite your best efforts to avoid it, I think I'd suggest letting it gradually warm to room temperature before trying to turn it on.