Can you choose a binary feature matrix for a binary classification model
I'm not aware of any literature specific to the case of classification based on binary features since it's just a subset of the general case, but it's definitely possible.
A very common example is traditional text classification, where the document is represented as a bag of words: there are different options but each word in the vocabulary can be represented as a boolean variable, representing whether it belongs to the document or not. For example (among many others), a Bernoulli Naive Bayes classifier can be trained on such data.