The border is a PDF annotation and as I saw in the hyperref code part of the hyperlink. Adding it without a link is seems to be not possible using existing LaTeX packages. The related pdfcomments package also provides some annotations but also no border only AFAIK. It might be possible using some direct PDF code using \pdfliteral, but you would need to know the PDF commands for this.
However you can add the border by placing it in a \hyperref or \href, but both do not allow paragraphs as content and generate a very tight border. You could use the v0.5 or higher of the adjustbox package (relative new) to box the minipage first which avoids the paragraph issue, add a margin if wanted and feed it to a \hyperref which e.g. simply points to the current page. The border color can be configured as described in the hyperref manual.
\documentclass{article}
\usepackage{hyperref}
\usepackage{adjustbox}
\newenvironment{hborderbox}[1]{%
\begin{adjustbox}{minipage=#1,margin=1ex,precode=\hyperlink{page:\the\numexpr\arabic{page}-1\relax}}%
}{%
\end{adjustbox}%
}
\usepackage{lipsum}
\begin{document}
\begin{hborderbox}{.5\textwidth}
\lipsum[1]\par\lipsum[2]
\end{hborderbox}
\end{document}
