Redefine \qedsymbol
:
\documentclass{article}
\usepackage{amsthm}
\usepackage{amssymb}
\renewcommand\qedsymbol{$\blacksquare$}
\begin{document}
\begin{proof}
A test text
\end{proof}
\end{document}

I wouldn't recommend to switch off right alignment for the end-mark (it's not standard, so readers might miss the end-mark and quite franckly, it looks ugly), but you could do something like this:
\documentclass{article}
\usepackage{amsthm}
\usepackage{amssymb}
\renewcommand\qedsymbol{$\blacksquare$}
\makeatletter
\renewenvironment{proof}[1][\proofname]{\par
\normalfont \topsep6\p@\@plus6\p@\relax
\trivlist
\item[\hskip\labelsep
\itshape
#1\@addpunct{.}]\ignorespaces
}{%
\nolinebreak\qedsymbol\endtrivlist\@endpefalse
}
\makeatother
\begin{document}
\begin{proof}
A test text some more text and some more text and some more text and
\end{proof}
\end{document}

As barbara beeton mentions in her comment
, it might be desired to turn off right alignment on a per-case basis; in this case, it's enough to locally redefine \qedsymbol
to be empty and then add the desired symbol at the end (taking some care to prevent undesired line breaks); something along these lines:
\documentclass{article}
\usepackage{amsthm}
\usepackage{amssymb}
\renewcommand\qedsymbol{$\blacksquare$}
\newcommand\placeqed{\nobreak\enspace$\blacksquare$}
\begin{document}
\begin{proof}
A test text some more text and some more text and some more text and
\end{proof}
\begin{proof}
\renewcommand\qedsymbol{}
A test text some more text and some more text and some more text and\placeqed
\end{proof}
\end{document}
