Bump right-aligned text to next line iff no room

You can encourage TeX not to break the lines with \linepenalty

\newcommand*{\FillLine}[2]{%
\noindent\parbox{\linewidth}{%
    \rightskip\fill\parfillskip-\rightskip
    \linepenalty100
    \exhyphenpenalty0
    File:~#2\linebreak[0] % <-- Need space here
    %Space before \hspace* allows for a break before it
    \hspace*{\fill}\texttt{\small CONFIDENTIAL}% 
}%
}

This is known in some inner circles as the Bourbaki problem. Here's an adaptation to your needs:

\documentclass{article}
\usepackage[paperwidth=7.0cm,showframe]{geometry}

\newcommand*{\FillLine}[2]{%
\noindent\parbox{\linewidth}{%
    \raggedright
    File:~#2%
    {\nobreak\hfill\penalty50\hskip1em\null\nobreak
     \hfill\texttt{\small CONFIDENTIAL}%
     \parfillskip=0pt \finalhyphendemerits=0 \par}
  }% End of \parbox
}

\begin{document}
\FillLine{Name}{Value}

\bigskip
\FillLine{Name}{A-Long-Value}

\bigskip
\FillLine{Name}{A-Long-Valu}

\bigskip
\FillLine{Name}{A-Long-Val}

\bigskip
\FillLine{Name}{A-Long-Va}

\bigskip
\FillLine{Name}{A-Long-V}

\bigskip
\FillLine{Name}{iiiiiii-iiiii-iiiiiii}

\bigskip
\FillLine{Name}{Some-Really-Really-Long-Value}
\end{document}

How does this work? Look for "Bourbaki" in the source file of the TeXbook.

I guess your friend is \looseness=-1: http://www.tug.org/utilities/plain/cseq.html#looseness-rp

It doesn't provide ideal result (the break in the last example gets modified), but it seems to work quite well.

\documentclass{article}
\usepackage[paperwidth=7.0cm,showframe]{geometry}
\usepackage{xcolor}
\usepackage{xparse}


\newcommand*{\FillLine}[2]{%
\noindent\parbox{\linewidth}{%
    \raggedright
    \parfillskip-\rightskip
    \looseness=-1
    File:~#2\ % <-- Need space here
    %Space before \hspace* allows for a break before it
    \hspace*{0pt plus 1fil}%
    \texttt{\small CONFIDENTIAL}% 
}%
}

%\newcommand*{\EnableHyphenationInTexttt}{\hyphenchar\font=45\relax}
%
%\newcommand*{\FillLine}[2]{%
%    \noindent\parbox{\linewidth}{%
%       \raggedright
%       File:~\texttt{\EnableHyphenationInTexttt#2} % <-- Need space here
%       %Space before \hspace* allows for a break before it
%       \hspace*{\fill}\texttt{\small CONFIDENTIAL}% 
%    }%
%}

\begin{document}
\FillLine{Name}{Value}

\bigskip
\FillLine{Name}{A-Long-Value}

\bigskip
\FillLine{Name}{iiiiiii-iiiii-iiiiiii}

\bigskip
\FillLine{Name}{Some-Really-Really-Long-Value}
\end{document}