You have a single class in the problem as you've posed it. This is not a one-vs.-all situation where you have multiple possible class outputs. If you were trying to classify pictures of cars into classes representing their body style, for example, you would need k output nodes (where k is the number of labeled classes of car bodies, e.g., sedan, pickup, hatchback, SUV, etc.). But all the training data must be labeled, and the output node that receives the largest output value determines the decision class, regardless of how close the others are. If you have more output nodes than you have labels the meaning of the NN is in question because you cannot train for the "none-of-the-above" label.
Supposing you were to have labels for "sedan," "hatchback," and "pickup" and then you wanted to add a catch-all category for none-of-the above. You would have to label all input images with one of the categories, even if the catch-all category has cars, kittens, kitchens, and koalas in it. The net will learn some feature set for the catch-all set, and what it actually learns is very difficult to determine. You will never really be able to fully train the catch-all set with a basis set of all possible other inputs. So the accuracy of its classification in this category is dubious. Worse, this additional category will dilute the other categories and increase the probability of false negatives in all the other classes.
So you are right to have a single output node with a single class. With NNs you have only known cases, there is no "else" catch-all block equivalent.