How to evaluate CCDE at a given frequency?
You need to follow the following steps (reponses at each steps are in a spoiler block, try to find them by yourself before checking). I will use the usual notation $z=e^{j w}$ for simplicity.


Find the relation between the transfer function $H(z)$ and the discrete-time Fourier Transform of the input $X(z)$ and the output $Y(z)$. This should be a basic results of your course


   $H(z) = \dfrac{Y(z)}{X(z)}$

Take the (discrete-time) Fourier Transform of your CCDE. For that, you will need to use two basic properties of the TF: linearity and time-shifting


   $Y(z) - \frac{1}{2} z^{-1} Y(z) = 2 X(z) - 5 z^{-1} X(z) - z^{-2} X(z)$

Compute $H(z)$ from the previous equation


   $H(z) = \dfrac{2 - 5z^{-1} - z^{-2}}{1-\frac{1}{2} z^{-1}}$

Finally take the value for $z = e^{j \pi} = -1$


   $H(e^{j \pi}) = \dfrac{2 + 5 - 1}{1+\frac{1}{2}} = \dfrac{6}{\frac{3}{2}} = 4$



This last result mean that your filter, at Nyquist frequency, has


   no phase delay and a gain of 4.