  Given a polynomial $f$ defined over a number field $K$, we make effective
certain special cases of a conjecture of S. Ih, on the finiteness of
$f$-preperiodic points which are $S$-integral with respect to a fixed
non-preperiodic point $\alpha$. As an application, we obtain bounds on the
number of $S$-units in the doubly indexed sequence $\{ f^n(\alpha) -
f^m(\alpha) \}_{n > m \geq 0}$. In the case of a unicritical polynomial
$f_c(z)=z^2+c$, with $\alpha$ fixed to be the critical point 0, for parameters
$c$ outside a small region, we give an explicit bound which depends only on the
number of places of bad reduction for $f_c$. As part of the proof, we obtain
novel lower bounds for the $v$-adically smallest preperiodic point of $f_c$ for
each place $v$ of $K$.
