Classification training data, but regression prediction
This is still a binary classification task.  In the abstract, there are two ways to handle this:


Most classifiers can output a predicted class and a confidence score (which indicates how confident the classifier is in its prediction).  If you don't need a probability, you can use the confidence score.  If you want to turn it into a probability, you can use various calibration procedures to turn this into a probability in the range [0,1].
Some classifiers can output a probability directly.  Generative models in particular are good at this.  For instance, logistic regression outputs both a predicted class and a probability for the class.


If you construct a neural network with $k$ outputs, one per class, using a softmax output and train it to minimize the cross-entropy loss function, then you can interpret the output as a probability distribution on the classes (though beware that it might be biased or over-confident).  The softmax ensures the outputs are normalized to be in the range [0,1] and to sum to one.

In your case, you can try both approaches (two outputs with softmax; or a single output with a sigmoidal activation function).  The only way to know which will perform better is to try both, but personally, I'd lean towards using two outputs and softmax.