Did you try to use the double $T$ of the surface $S$? Any fixed point-free action of $\mathbf Z/n$ on $S$ induces a fixed point-free action on the closed orientable surface $T$. Moreover, the induced action commutes with the natural orientation-reversing involution on $T$. By Nielsen's Theorem that you mentioned, you've reduced to study orientation-reversing involutions on $T$ that commute with a given action of $\mathbf Z/n$ on $T$ and whose set of fixed points is homeomorphic to a disjoint union of $b$ circles. Such involutions induce orientation-reversing involutions on the quotient surface $U=T/(\mathbf Z/n)$. Now, orientation-reversing involutions on a closed orientable surface $U$ whose set of fixed points is a disjoint union of circles are easily classified. It should not be too difficult to decide which ones lift to orientation-reversing involutions on $T$ having a set of fixed points composed of $b$ circles.

With this approach you already get conditions on the mere existence of fixed point-free actions of $\mathbf Z/n$ on $S$. Indeed, the double $T$ of $S$ has genus $2g+b-1$. So that $n$ has to divide
$$
-\chi(T)=2(2g+b-1)-2=4g+2b-4
$$
in order for a fixed point-free action of $\mathbf Z/n$ on $S$ to exist. For example, if $S$ is a pair of pants, i.e., $g=0$ and $b=3$, the natural number $n$ has to divide $2$. Adopting the above strategy, I think you only get $1$ fixed point free action of $\mathbf Z/2$ on the pair of pants $S$: the one where the nontrivial element of $\mathbf Z/2$ acts on $S$ by exchanging $2$ boundary circles, and by acting antipodally on the third. That case probably was clear anyway.