= Solution
Use the four supplied opposite-site combinations in the order $(\phi_1,\phi_2,\phi_3,\phi_4)$. Their <Hückel method> <matrix> is
$$
H_{E_g}=\begin{pmatrix}
\alpha&\beta_S&0&-\beta_L\\
\beta_S&\alpha&\beta_L&0\\
0&\beta_L&\alpha&\beta_S\\
-\beta_L&0&\beta_S&\alpha
\end{pmatrix}.
$$
The negative corner is essential. For example, $\langle\phi_4|H|\phi_1\rangle$ includes the long bonds $4$–$5$ and $8$–$1$, each with opposite coefficient signs, giving $-\beta_L$.
Because the basis is orthonormal, the secular <determinant> is
$$
\boxed{\begin{vmatrix}
\alpha-E&\beta_S&0&-\beta_L\\
\beta_S&\alpha-E&\beta_L&0\\
0&\beta_L&\alpha-E&\beta_S\\
-\beta_L&0&\beta_S&\alpha-E
\end{vmatrix}=0.}
$$
It factors as $\big[(\alpha-E)^2-\beta_S^2-\beta_L^2\big]^2=0$, giving two doubly degenerate <energies>, $\alpha\pm\sqrt{\beta_S^2+\beta_L^2}$.
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