squote.tex

\documentclass{article}
\usepackage{textcomp}

\makeatletter
\let\orgtexttt\texttt
\edef\x@store{\the\catcode`\'}
\catcode`\'=\active
\def\texttt{%
  \bgroup
  \catcode`\'=\active
  \x@texttt}
\def\x@texttt#1{%
  \def'{\textquotesingle}%
  \orgtexttt{#1}\egroup}
\catcode`\'=\x@store
\makeatother

\begin{document}

$a'$ said \texttt{'Hi'}.

\end{document}