In this paper, a general binary-input binary-output (BIBO) channel is investigated in the presence of feedback and input constraints. The feedback capacity and the optimal input distribution of this setting are calculated for the case of an $(1,\infty)$-RLL input constraint, that is, the input sequence contains no consecutive ones. These results are obtained via explicit solution of a corresponding dynamic programming optimization problem. A simple coding scheme is designed based on the principle of posterior matching, which was introduced by Shayevitz and Feder for memoryless channels. The posterior matching scheme for our input-constrained setting is shown to achieve capacity using two new ideas: \textit{message history}, which captures the memory embedded in the setting, and \textit{message splitting}, which eases the analysis of the scheme. Additionally, in the special case of an S-channel, we give a very simple zero-error coding scheme that is shown to achieve capacity. For the input-constrained BSC, we show using our capacity formula that feedback increases capacity when the cross-over probability is small.