How binary quantile regression divides the dependent variable into quantiles I'm not very clear with binary quantile regression. As if it was ordinary quantile regression, it would divide the dependent variable's value by its ascending value into quantiles.
But I cannot imagine how it divides $y \{0;1\}$ value into quantiles. Can you explain it to me?
 A: Binary quantile regression does not actually classify 

It classifies some 

I.e. the (predicted) probability that the target variable (regressand, dependent variable) is 1 (or alternatively 0) based on the input variable(s) (regressor(s), independent variable(s)). Details depend greatly on the method, but one can see how a median predicted value of the variable would allow for a quantile distribution.
Further Reading
Intro To Binary Quantile Regression Discussion
Recent Binary Quantile Regression Method Research
A: I don't think that "binary" and "quantile" should be used in the same sentence.  Quantile regression requires not only multiple levels of $Y$ but that $Y$ be continuous.  Sticking with ordinary logistic regression is more appropriate.
A: Imagine an unobserved random variable Z with a logistic density with mean E[Z| X] where X may be a vector of covariates.  We may ask for a predictor of the quantile of Z given predictors X.
A: Quantiles are invariant to monotone transformations. Binary quantile regression uses the fact that the indicator function is monotone. You actually model the conditional quantiles of the underlying latent continuous variable of which the observed binary is just an indicator.
