what does \$X(z) \pm X(-z)\$ mean?
If you want to know what happens to the frequency spectrum, you need to evaluate the output on the unit circle \$z=e^{j\omega}\$. Since \$-z=-e^{j\omega}=e^{j(\omega+\pi)}\$, the output spectrum is

$$Y(e^{j\omega})=X(e^{j\omega})+X(e^{j(\omega+\pi)})$$

That means that the output spectrum is the sum of the original input spectrum and a shifted version of the input spectrum, where the amount of shift equals \$\pi\$ (i.e. half the sampling frequency).