With the newest version of adjustbox v1.2 from 2019/01/04 you can use the pagecenter key to align the content centered on the page. This uses the code in my original answer internally (i.e. lapping the content to the left depending on the page margin).
\documentclass[twopages]{book}
\usepackage{adjustbox}[2019/01/04]
\usepackage{blindtext}
\begin{document}
\blindtext
\begin{adjustbox}{minipage=0.95\paperwidth,frame,pagecenter,vspace=\medskipamount}
\blindtext
\end{adjustbox}
\blindtext
\clearpage
\blindtext
\begin{adjustbox}{minipage=0.95\paperwidth,frame,pagecenter,vspace=\medskipamount}
\blindtext
\end{adjustbox}
\blindtext
\end{document}
Original answer which works with older versions of adjustbox:
If you want to center the box on the page, not the text area, one way to do this is to make it lap into the left text margin of the page. The required dimensions can be read using the layout package.
The following code uses the ifoddpage package together with adjustbox to get the right amount.
\documentclass[twopages]{book}
\usepackage{ifoddpage}
\usepackage{adjustbox}
\usepackage[pass,showframe]{geometry}
\usepackage{layout}
\usepackage{lipsum}
\begin{document}
\lipsum[1]
\checkoddpage
\begin{adjustbox}{minipage=0.95\paperwidth,vspace=\smallskipamount,frame,
lap={\textwidth}{-1in-\hoffset-\ifoddpage\oddsidemargin\else\evensidemargin\fi+.5\paperwidth-.5\width-\csname @totalleftmargin\endcsname}}
\lipsum[2]
\end{adjustbox}
\lipsum
\checkoddpage
\begin{adjustbox}{minipage=0.95\paperwidth,vspace=\smallskipamount,frame,
lap={\textwidth}{-1in-\hoffset-\ifoddpage\oddsidemargin\else\evensidemargin\fi+.5\paperwidth-.5\width-\csname @totalleftmargin\endcsname}}
\lipsum[2]
\end{adjustbox}
\lipsum[1]
\clearpage
\layout
\end{document}


\textwidth– barbara beeton Jun 09 '18 at 19:57