The answer is yes.

Let's prove this by induction on $n$. If $n=1$, then by M. Hall's theorem there is a finite index subgroup $K \leqslant F$ such that $H_1 \subseteq K$ and $|F:K|>1/\epsilon$. Choose any finite index normal subgroup $N \lhd F$, which is contained in $K$, let $G=F/N$ and let $\phi:F \to G$ be the natural epimorphism. Note that $|G:\phi(K)|=|F:K|$ as $N\subseteq K$. Then
$$\frac{|\phi(H_1)|}{|G|} \le \frac{|\phi(K)|}{|G|}=\frac{1}{|G:\phi(K)|}=
\frac{1}{|F:K|}<\epsilon .$$

Now, suppose that $n\ge 2$ and the statement has been proved for smaller $n$. By Theorem 1.1 in http://arxiv.org/abs/1308.3192, there is a finitely generated subgroup $H$ of infinite index in $F$, such that $H_{n-1} \subseteq H$ and $|H_n: (H_n \cap H)|=k<\infty$.

Thus $H_n \subseteq \bigcup_{j=1}^k H f_j$ for some $f_1,\dots, f_k \in F$. By the induction hypothesis, we can find an epimorphism $\phi$ from $F$ onto a finite group $G$ such that $$\frac{|\phi(H_1\dots H_{n-2}H)|}{|G|}< \frac\epsilon k.$$
Since $H_1 \dots H_n \subseteq \bigcup_{j=1}^k H_1 \dots H_{n-2}Hf_j$, we have
\begin{multline*}\frac{|\phi(H_1\dots H_{n-2}H_{n-1}H_n)|}{|G|}\le \sum_{j=1}^k \frac{|\phi(H_1\dots H_{n-2}H)\phi(f_j)|}{|G|} \\ =\sum_{j=1}^k \frac{|\phi(H_1\dots H_{n-2}H)|}{|G|}<k \, \frac\epsilon k =\epsilon,\end{multline*} as required.