You CANNOT use fmincon on a problem with binary variables or any form of discrete variables.
However, IF y is indeed binary, then you have only two cases to consider. So solve the problem to minimize f(x), given y == 0, and then repeat, solving it for the minimum over x of f(x), given y == 1.
Take the better of the two results and you are done. There really is little more than that to do here. You still need to choose intelligent starting values for x of course. Note that if there are multiple disjoint regions for x that satisfy the constraints, thus g(x,y)<=0, then you need to search within EACH of them. Fmincon cannot intelligently jump from one such region to another to search among them all.
If you had a more complicated case where you had multiple binary variables, then you would be forced to use a code like GA, which can handle the fully general problem. But that is apparently not the case here.