converting block diagrams with delays to equations
If you are confused about the interpretation, why not divide the diagram into 2 parts. The signal to the left of the rightmost delay element is assumed to be z[n].  

Thus, now $$y[n] = z[n-1]$$  

$$z[n] = x[n] + (-1)\cdot x[n-1]$$  

Thus

$$y[n] = x[n-1]  -x[n-2]$$