Specific \epigraph style

All you need to add to your code is

\setlength\epigraphrule{0pt}

A complete example:

\documentclass{book}
\usepackage{epigraph}

\setlength\epigraphwidth{.8\textwidth}
\setlength\epigraphrule{0pt}

\begin{document}

\thispagestyle{empty}
\vspace*{\fill}
\epigraph{\itshape Begin at the beginning, the King said gravely, ``and go on till you come to the end: then stop.''}{---Lewis Carroll, \textit{Alice in Wonderland}}

\end{document}

A solution consists in patching some epigraph internal commands, with the etoolbox package, so that the default font shape for the epigraph text be italic (in an upshape context):

    \documentclass[12pt]{book}
    \usepackage[utf8]{inputenc}
    \usepackage[T1]{fontenc}
    \usepackage{amsmath,amsfonts,mathabx}
    \usepackage{ebgaramond}
    \usepackage{epigraph}
    \renewcommand\textflush{flushright}

    \usepackage{etoolbox}
    \makeatletter
    \newlength\epitextskip
    \pretocmd{\@epitext}{\em}{}{}
    \apptocmd{\@epitext}{\em}{}{}
    \patchcmd{\epigraph}{\@epitext{#1}\\}{\@epitext{#1}\\[\epitextskip]}{}{}
    \makeatother

    \setlength\epigraphrule{0pt}
    \setlength\epitextskip{2ex}
    \setlength\epigraphwidth{.8\textwidth}

    \thispagestyle{empty}

    \begin{document}

    \vspace*{\fill}
    \epigraph{For the Snark \emph{was} a Boojum,  you see.}{Lewis Carroll, \emph{The Hunting of the Snark}
    }

    \end{document} 

Not the answer to your question, but a solution with KOMA-Script.

Option 1:

\documentclass{scrbook}

\setkomafont{dictumtext}{\itshape\small}
\setkomafont{dictumauthor}{\normalfont}
\renewcommand*\dictumwidth{.75\linewidth}
\renewcommand*\dictumauthorformat[1]{--- #1}
\renewcommand*\dictumrule{}

\begin{document}

\thispagestyle{empty}
\vspace*{\fill}
\dictum[Lewis Carroll, \emph{Alice in Wonderland}]{Begin at the beginning, the King said gravely, ``and go on till you come to the end: then stop.''}

\end{document}

Option 2:

\documentclass{book}
\usepackage{scrextend}

\setkomafont{dictumtext}{\itshape\small}
\setkomafont{dictumauthor}{\normalfont}
\renewcommand*\dictumwidth{.75\linewidth}
\renewcommand*\dictumauthorformat[1]{--- #1}
\renewcommand*\dictumrule{}

\begin{document}

\thispagestyle{empty}
\vspace*{\fill}
\dictum[Lewis Carroll, \emph{Alice in Wonderland}]{Begin at the beginning, the King said gravely, ``and go on till you come to the end: then stop.''}

\end{document}

Option 3:

\documentclass{book}

\makeatletter
\newcommand{\dictum}[2][]{\par%
  \begingroup
    \raggeddictum\parbox{\dictumwidth}{%
      {\@dictumfont{\raggeddictumtext #2\strut\par}%
        \def\@tempa{#1}\ifx\@tempa\@empty\else%
          {\raggeddictumtext\dictumrule}%
          \raggeddictumauthor\@dictumauthorfont\dictumauthorformat{#1}%
          \strut\par%
        \fi%
      }%
    }\par%
  \endgroup
}

\newcommand*{\dictumrule}{}
\newcommand*{\dictumwidth}{.75\textwidth}
\newcommand*{\raggeddictum}{\raggedleft}
\newcommand*{\dictumauthorformat}[1]{--- #1}
\newcommand*{\raggeddictumtext}{\raggedright}
\newcommand*{\raggeddictumauthor}{\raggedleft}
\newcommand*{\@dictumfont}{\itshape\small}
\newcommand*{\@dictumauthorfont}{\normalfont}
\makeatother

\begin{document}

\thispagestyle{empty}
\vspace*{\fill}
\dictum[Lewis Carroll, \emph{Alice in Wonderland}]{Begin at the beginning, the King said gravely, ``and go on till you come to the end: then stop.''}

\end{document}