= Solution
For the $p_z$ <atomic orbital> basis, only orbitals whose centres remain fixed contribute to a <character>. Each such orbital contributes $+1$ if its $z$ direction is preserved and $-1$ if it is reversed. In the standard <D4h molecular point group> class order,
$$
\begin{array}{c|rrrrrrrrrr}
&E&2C_4&C_2&2C_2'&2C_2''&i&2S_4&\sigma_h&2\sigma_v&2\sigma_d\\\hline
\chi_{p_z}&8&0&0&0&0&0&0&-8&0&0
\end{array}
$$
No nonidentity operation other than $\sigma_h$ leaves any <carbon> site fixed. Reflection in the molecular plane fixes all eight sites but reverses every $p_z$ function.
<Molecular character reduction> gives
$$
\boxed{\Gamma_{p_z}=A_{1u}+A_{2u}+B_{1u}+B_{2u}+2E_g.}
$$
The dimensions sum to $1+1+1+1+2\cdot2=8$. The two copies of $E_g$ are separate equivalent <irreducible representations> which may mix with one another; they do not force all four functions to have a common <energy>.
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