Is a Neural Network with 20 times the number of input neurons (on hidden layers) guaranteed to overfit? When is this not so?
Figuring out the correct size for hidden layers in NN is a bit of an art, that said I have never used hidden layers with more than twice the number of neurones as the input layer.

As an example the MNIST problem has an input size of 784, I cannot imagine how the network will not overfit if the hidden layer is of size 784*20...

So answering to your question, I think overfit will pretty much occur on those occasions, if unsure, just try it in the MNIST dataser