\documentclass{article}

\usepackage{environ,newfile,etoolbox}

\newoutputstream{verbatimcontent}

\NewEnviron{myitemize}[2]{%
  % Find #1 and replace with #2
  % \patchcmd{<cmd>}{<search>}{<replace>}{<success>}{<failure>}
  \patchcmd{\BODY}{#1}{#2}{}{}%
  \openoutputfile{tempfile.txt}{verbatimcontent}
  {\let\item\relax
   \let\oldverb\relax
   \def\verb{\oldverb{}}
  \addtostream{verbatimcontent}{\BODY}}
  \closeoutputstream{verbatimcontent}
  \def\oldverb##1{\verb}%
  \begin{itemize}
    \input{tempfile.txt}
  \end{itemize}
}

\begin{document}

\begin{itemize}
  \item \verb|test|
\end{itemize}

\begin{myitemize}{test}{testing}
  \item \verb|test &^!\LaTeX|
\end{myitemize}

\end{document}