The naswer is no. For example, the 4-by-4 matrix $\begin{pmatrix} p & e \\ f & q \end{pmatrix}$, with $p=\begin{pmatrix} 1 & 0\\ 0 & 0\end{pmatrix}$, $q=1-p$, $e=\begin{pmatrix} 1/2 & 1/2\\ 1/2 & 1/2\end{pmatrix}$, and $f=1-e$ has the characteristic polynomial $x^4-2x^3+x^2+1/4=(x^2-x)^2+1/4\geq1/4$ and has no real eigenvalues.

The answer is no. For example, the 4-by-4 matrix $\begin{pmatrix} p & e \\ f & q \end{pmatrix}$, with $p=\begin{pmatrix} 1 & 0\\ 0 & 0\end{pmatrix}$, $q=1-p$, $e=\begin{pmatrix} 1/2 & 1/2\\ 1/2 & 1/2\end{pmatrix}$, and $f=1-e$ has the characteristic polynomial $x^4-2x^3+x^2+1/4=(x^2-x)^2+1/4\geq1/4$ and has no real eigenvalues.