= Solution
A <linear combination of atomic orbitals> expands a <crystal> wavefunction in localized orbitals on all sites,
$$
\Psi_{\mathbf k}(\mathbf r)=\sum_{\mathbf R,\alpha,\eta}c_{\alpha\eta}(\mathbf k)
e^{i\mathbf k\cdot(\mathbf R+\mathbf d_\alpha)}
\varphi_{\alpha\eta}(\mathbf r-\mathbf R-\mathbf d_\alpha).
$$
Here $\alpha$ labels the <crystal motif>[basis] site, $\eta$ the retained <atomic orbitals> and $\mathbf R=m\mathbf a_1+n\mathbf a_2$. For the two-band model retain one relevant orbital on each B or N atom, usually the out-of-plane $p_z$ orbital, and neglect other orbital manifolds. Define normalized Bloch sums for $N_c$ cells,
$$
\Phi_\alpha(\mathbf k,\mathbf r)=\frac1{\sqrt{N_c}}\sum_{m,n}
e^{i\mathbf k\cdot(m\mathbf a_1+n\mathbf a_2+\mathbf d_\alpha)}
\varphi_\alpha(\mathbf r-m\mathbf a_1-n\mathbf a_2-\mathbf d_\alpha),
\quad\alpha=B,N.
$$
Then $\Psi=\alpha_B\Phi_B+\alpha_N\Phi_N$. Bloch periodicity puts the same two coefficients in every cell. We adopt the stated <orthonormality>, neglect longer-range hopping, and approximate the diagonal <matrix> elements by the supplied atomic energies $\epsilon_B,\epsilon_N$. More generally the <crystal> environment shifts these on-site energies, so equality with isolated-atom expectation values is an additional <tight-binding approximation>.
For equal real nearest-neighbour transfer <integral> $t_{\rm nn}$, this phase convention gives
$$
\gamma(\mathbf k)=\sum_{j=1}^3e^{i\mathbf k\cdot\mathbf n_j},\qquad
H(\mathbf k)=\begin{pmatrix}\epsilon_B&t_{\rm nn}\gamma\\t_{\rm nn}\gamma^*&\epsilon_N\end{pmatrix}.
$$
The secular determinant is $(\epsilon_B-E)(\epsilon_N-E)-t_{\rm nn}^2|\gamma|^2=0$. Thus
$$
\boxed{E_\pm(\mathbf k)=\frac{\epsilon_B+\epsilon_N}{2}
\pm\sqrt{\frac{(\epsilon_B-\epsilon_N)^2}{4}+t_{\rm nn}^2|\gamma|^2}.}
$$
Since $(\epsilon_B-\epsilon_N)^2/4=(\epsilon_B+\epsilon_N)^2/4-\epsilon_B\epsilon_N$, this is the requested algebraic form. A hopping-sign convention or rephasing of the Bloch sums changes off-diagonal phases but not these <eigenvalues>. Explicitly,
$$
\gamma=e^{ik_xa/\sqrt3}+2e^{-ik_xa/(2\sqrt3)}\cos(k_ya/2),\quad
|\gamma|^2=1+4\cos^2(k_ya/2)+4\cos(k_ya/2)\cos(\sqrt3ak_x/2).
$$
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