$\newcommand{\from}{\colon}\newcommand{\Aut}{\rm{Aut}}\newcommand{\bdy}{\partial}$Here is a fairly hands-on proof. I'll use the following notation: $H = H_g$, $S = S_g = \bdy H$, and $F = F_g$. I'll write $g(S) = g(H) = g$. Also, $i \from S \to H$ is the inclusion map.
Suppose that $f_* \from \pi_1(S) \to F = \pi_1(H)$ is an epimorphism. Then there is a homeomorphism $k \from S \to S$ so that $i_* \circ k_* = f_*$.
Fact number one: since $S$ and $H$ are Ellenberg-Maclane spaces, there is a map $f \from S \to H$ inducing $f_*$. Fix a cut system $D \subset H$: that is, a collection of $g(H)$ disks so that $H - n(D)$ is a three-ball. Homotope $f$ to make it transverse to $D$. Thus $C = f^{-1}(D)$ is a collection of simple closed curves in $S$. For any $\alpha \subset C$, let $D(\alpha)$ be the disk containing $f(\alpha)$.
We now construct a graph $G$. We have a vertex $v(X)$ for every component $X$ of $S - C$. We have an edge $e(\alpha)$ for every component $\alpha$ of $C$. We connect one (both) end(s) of $e(\alpha)$ to $v(X)$ if one (both) side(s) of $\alpha$ are contained in $X$. Note that $G$ is a finite connected graph.
Recall that the genus $g(S)$ of $S$ is the number of curves we must cut along to obtain a connected planar surface. In similar fashion we define the genus $g(G')$ of $G$ to be the number of edges we must remove to obtain a tree. We deduce that $g(G) \leq g(S)$. The retraction $r \from S \to G$ induces an epimorphism $r_* \from \pi_1(S) \to \pi_1(G)$, a free group of rank $g(G)$. Fix a point $x \in H - n(C)$. Define a map $h \from G \to H$ as follows. All vertices are mapped to $x$. The edge $e(\alpha)$ is sent to any loop based at $x$ and meeting $C$ exactly once, transversely, at a point of $D(\alpha)$. (I am ignoring a small issue about orientations of edges here.) We deduce that $f_* = h_* \circ r_*$. Thus $h_* \from \pi_1(G) \to F$ is an epimorphism. Fact number two: since free groups are Hopfian, $g(G) = g(S)$ and $h_*$ is an isomorphism.
We now construct a handlebody $H'$, homotopy equivalent to $G$. Take $S$, thicken to get $S \times [0,1]$, and attach two-handles along the curves $C \times \{0\}$. We next attach three-handles to the sphere components of the lower boundary. We identify the groups $\pi_1(H')$ and $\pi_1(G)$. Let $i' \from S \to H'$ be the inclusion map.
Fact number three: using Nielsen's theorem (giving generators of $\Aut(F)$), there is a homeomorphism $h' \from H' \to H$ inducing $h_*$. Consider the homeomorphism $k = (\bdy i)^{-1} \circ \bdy h' \circ \bdy i'$, where all domains and ranges are restricted to be $S$. Chasing the diagram of spaces shows that $i_* \circ k_* = f_*$, and we are done.