Quantum tunnelling is more that a particle, such as an electron, has a certain probability of being on the other side of a potential barrier. It is like a car on one side of a mountain, but without enough fuel to get to the top and over the mountain, can sometimes appear at a similar altitude on the other side of the mountain. It is as thoughit had "tunnelled" through it. And it's potential energy does not need to change. This is due to a particle not behaving like a, more familiar, car but having associated with it only a probability of position. This positional probability is spread over the whole universe but just weighted towards the "traditional" view.

There is no model, that I know of, for the view that such a probability could extend beyond the known universe. By definition of the word "universe" this cannot happen. But one can imagine a multiverse of self contained black holes so it is certainly possible to "think" of an outside view. Remember this is only one of many ideas and we have no way of knowing whether such a concept is valid or not. I don't think we have got so far with tying together quantum mechanics and cosmology to have any sort of model that relates the probability amplitude of an electron to the size of the universe (or knowable universe) but I expect there may well be some relationship.

Your questions go very deep and it is currently beyond human knowledge to answer I think. Certainly beyond mine :-)