How do I create this semi-enclosed arrow symbol?

Welcome! You do not need any packages for that.

\documentclass{article}
\newcommand{\RightArrowInBox}{\begingroup
\renewcommand{\arraystretch}{0.6}\begin{array}{@{}c@{}|}\hline 
\raisebox{-0.2ex}{$\Rightarrow$}\\ \hline\end{array}\endgroup}
\newcommand{\LeftArrowInBox}{\begingroup
\renewcommand{\arraystretch}{0.6}\begin{array}{@{}c@{}|}\hline 
\raisebox{-0.2ex}{$\Leftarrow$}\\ \hline\end{array}\endgroup}
\begin{document}
$\RightArrowInBox~\LeftArrowInBox$
\end{document}

Of course you can modify the padding etc.

A version without cutting corners with major input by barbara beeton.

\documentclass{article}
\newcommand{\RightArrowInBox}{\begingroup
\renewcommand{\arraystretch}{0.6}\begin{array}{@{}c@{\rule[-3pt]{0.4pt}{0.85em}}}\hline 
\raisebox{-0.28ex}{$\Rightarrow$}\\ \hline\end{array}\endgroup}
\newcommand{\LeftArrowInBox}{\begingroup
\renewcommand{\arraystretch}{0.6}\begin{array}{@{}c@{\rule[-3pt]{0.4pt}{0.85em}}}\hline 
\raisebox{-0.28ex}{$\Leftarrow$}\\ \hline\end{array}\endgroup}
\begin{document}
$\RightArrowInBox~\LeftArrowInBox$
\end{document}

Or with trimclip.

\documentclass{article}
\usepackage{trimclip}
\newcommand{\RightArrowInBox}{\begingroup
\clipbox{0.5ex 0em 0em 0em}{\fbox{$\Rightarrow\!$}}\endgroup}
\newcommand{\LeftArrowInBox}{\begingroup
\clipbox{0.5ex 0em 0em 0em}{\fbox{$\Rightarrow\!$}}\endgroup}

\begin{document}
$\RightArrowInBox~\LeftArrowInBox$
\end{document}

Here's a solution which places \implies and \impliedby into a modified version of \fbox.

\documentclass{article}
\usepackage{amsmath}
\makeatletter
\def\@rframeb@x#1{%
  \@tempdima\fboxrule
  %\advance\@tempdima\fboxsep
  \advance\@tempdima\dp\@tempboxa
  \hbox{%
    \lower\@tempdima\hbox{%
      \vbox{%
        \hrule\@height\fboxrule
        \hbox{%
          %\vrule\@width\fboxrule
          #1%
          \vbox{%
            \vskip.7\fboxsep
            \box\@tempboxa
            \vskip.3\fboxsep}%
          #1%
          \vrule\@width\fboxrule}%
        \hrule\@height\fboxrule}%
                          }%
        }%
}
\DeclareRobustCommand\rfbox[1]{%
  \leavevmode
  \setbox\@tempboxa\hbox{%
    \color@begingroup
      \kern.2\fboxsep{#1}\kern.2\fboxsep
    \color@endgroup}%
  \@rframeb@x\relax}
\makeatother
\newcommand{\boximplies}{%
    \mathrel{\:\rfbox{$\Longrightarrow$}\:}%
}
\newcommand{\boximpliedby}{%
    \mathrel{\:\rfbox{$\Longleftarrow$}\:}%
}
\begin{document}
\( A \implies B \)

\( A \boximplies B \)

\( A \impliedby B \)

\( A \boximpliedby B \)
\end{document}

Only array (that's a required package in every distribution):

\documentclass{article}
\usepackage{array}

\newcommand{\bimplies}{\mathrel{\bimpl{c|}{\Rightarrow}}}
\newcommand{\bimpliedby}{\mathrel{\bimpl{|c}{\Leftarrow}}}
\newcommand{\bimpl}[2]{%
  \begingroup
  \setlength{\arraycolsep}{0pt}%
  \renewcommand{\arraystretch}{0}%
  \begin{array}{#1}\hline{#2}\vphantom{+}\\\hline\end{array}%
  \endgroup
}

\begin{document}

$A\bimplies B\bimpliedby C$

\end{document}