Misalignment of \vrule

The fixed 3in rule width is good, but is also the problem here. Perhaps you can adjust your \rightline usage by using a fixed-width box (something like tabular). This way contents below the signature line will wrap as needed. I've defined \signatureline to be of width .6\linewidth, pushing its contents flush right as needed:

\documentclass{article}
\newcommand{\signatureline}[1]{%
  \begin{tabular}{@{}p{.6\linewidth}@{}}
    \hline \\[-.5\normalbaselineskip] #1
  \end{tabular}
}
\begin{document}
\noindent Approved:\par%
\vspace{0.15in}
\null \hfill \signatureline{Advisor\\A.\ B.\ Coverage\\Dept.\ of Electrical and Computer Engineering}
\vfill

\noindent Committee Members:\par%
\vspace{0.15in}
\noindent \hfill \signatureline{First Reader\\Dept.\ of Electrical and Computer Engineering}
\vfill

\noindent \hfill \signatureline{Second Reader\\Dept.\ of Physics}
\vfill

\end{document}

I've kept the other spacing as-is, although I would adjust them as well.

The \rightline macro will shove its contents to the right of the page. So your problem is not that the second reader's line is moved too far to the right, but that your first reader, and your committee chair, are pushed left a bit because their departmental affiliation lines are more than 3in long. You can either modify the template so that the \vrule width 3in is replaced with a sufficiently wide \vrule (determined by trial and error) or you can insert a manual line break in the affiliation text, e.g. Dept. of Electrical and\\Computer Engineering. Or use abbreviations, I suppose, depending on whether that is okay.