The solutions to $z^4 = 4 - 4i \sqrt{3}$ can be expressed in the form
\begin{align*}
z_1 &= r_1 (\cos \theta_1 + i \sin \theta_1), \\
z_2 &= r_2 (\cos \theta_2 + i \sin \theta_2), \\
z_3 &= r_3 (\cos \theta_3 + i \sin \theta_3), \\
z_4 &= r_4 (\cos \theta_4 + i \sin \theta_4),
\end{align*}where $r_k > 0$ and $0^\circ \le \theta_k < 360^\circ.$  Find $\theta_1 + \theta_2 + \theta_3 + \theta_4,$ in degrees.
First, we can write $z^4 = 4 - 4i \sqrt{3} = 8 \operatorname{cis} 300^\circ.$  Therefore, the four roots are
\begin{align*}
&\sqrt[4]{8} \operatorname{cis} 75^\circ, \\
&\sqrt[4]{8} \operatorname{cis} (75^\circ + 90^\circ) = \sqrt[4]{8} \operatorname{cis} 165^\circ, \\
&\sqrt[4]{8} \operatorname{cis} (75^\circ + 180^\circ) = \sqrt[4]{8} \operatorname{cis} 255^\circ, \\
&\sqrt[4]{8} \operatorname{cis} (75^\circ + 270^\circ) = \sqrt[4]{8} \operatorname{cis} 345^\circ.
\end{align*}Then $\theta_1 + \theta_2 + \theta_3 + \theta_4 = 75^\circ + 165^\circ + 255^\circ + 345^\circ = \boxed{840^\circ}.$