\hrulefill suddenly disappearing

blockquote does not indent its argument text, so there's no need for \noindent. However, to provide a full line of fill, set something (nothing) at the beginning of the line:

\documentclass[12pt]{article}
\usepackage[a4paper]{geometry}
\usepackage{csquotes,lipsum}

\begin{document}

\lipsum[1]

\blockquote{\mbox{}\hrulefill \par
Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi. 
Morbi auctor lorem non justo. Nam lacus libero, pretium at, lobortis 
vitae, ultricies et, tellus. Donec aliquet, tortor sed accumsan 
bibendum, erat ligula aliquet magna, vitae ornare odio metus a mi. 
Morbi ac orci et nisl hendrerit mollis. Suspendisse ut massa. Cras 
nec ante. Pellentesque a nulla. Cum sociis natoque penatibus et 
magnis dis parturient montes, nascetur ridiculus mus. Aliquam 
tincidunt urna. Nulla ullamcorper vestibulum turpis. Pellentesque 
cursus luctus mauris. 
\\\mbox{}\hrulefill}

\lipsum[3]

\end{document}

If this type of decoration is something you're interested in using more frequently, you could incorporate it in a fixed command, rather than adding horizontal rules manually. It provides consistency. Below is an example that uses a regular tabular to set the quote (foregoing page breaking, of course):

\documentclass[12pt]{article}
\usepackage[a4paper]{geometry}
\usepackage{lipsum}

\newcommand{\blockquote}[1]{\par\bigskip{%
  \centering\begin{tabular}{p{.85\linewidth}}
    \hline \\[\dimexpr-\normalbaselineskip+5pt] #1 \\[5pt] \hline
  \end{tabular}\par}\bigskip}
\begin{document}

\lipsum[1]

\blockquote{%
Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi. 
Morbi auctor lorem non justo. Nam lacus libero, pretium at, lobortis 
vitae, ultricies et, tellus. Donec aliquet, tortor sed accumsan 
bibendum, erat ligula aliquet magna, vitae ornare odio metus a mi. 
Morbi ac orci et nisl hendrerit mollis. Suspendisse ut massa. Cras 
nec ante. Pellentesque a nulla. Cum sociis natoque penatibus et 
magnis dis parturient montes, nascetur ridiculus mus. Aliquam 
tincidunt urna. Nulla ullamcorper vestibulum turpis. Pellentesque 
cursus luctus mauris.}

\lipsum[3]

\end{document}

To allow page breaking of such quotes, one could also consider mdframed.