Remove the call of the breqn
package.
After the operation I get

Note that subfigure
has been obsolete for several years. The official replacement is subfig
. Alternatively, use subcaption
. Note that the class already loads caption
, so subcaption
should probably be chosen.
I add a version without \left
and right
, which don't seem necessary (they rarely are in cases like this) and where the superscripts (n) are similar to the one for Gamma. More importantly, the two consecutive equations are together in a gather
environment: two equation
environments should never appear immediately after one another.
\begin{subequations}
\begin{gather}
[ \tilde{V}^{-^{(n)}} ] = [ \Gamma^{(n)} ] [ \tilde{V}^{+^{(n)}} ],
\\
[ \tilde{V}^{-\,(n)} ] = [ \Gamma^{(n)} ] [ \tilde{V}^{+\,(n)} ],
\end{gather}
\end{subequations}

In order to accommodate the split formula where you use \left
and \right
across lines, just remove all of them; for better clarity, some of the fences can be made \big
, but not more than that. Note that multline
is the environment of choice for this.
\begin{multline}
[V_m^{(n)}(z)]=[ P^{(n)} ]
\bigl( \bigl[ \exp \bigl( -j k ^{(n)} ( z-z_n) \bigr) \bigr] [ \tilde{V}^{+^{(n)}}] + \\
\bigl[ \exp \bigl( +j k ^{(n)} ( z-z_{n+1} ) \bigr) \bigr] [ \tilde{V}^{-^{(n)}} ]\bigr)
\end{multline}

That's the chi in the Symbol Adobe font.
\documentclass{article}
\usepackage{mathptmx}
\usepackage{amsmath}
\newcommand{\upchi}{\text{\usefont{U}{psy}{m}{n}\symbol{'143}}}
\begin{document}
X\upchi
$\upchi+X_{\upchi}$
\end{document}

Best Answer
since many of the greek uppercase letters have the same shape as latin letters, they were not separately provided for tex, for two reasons:
to save space (which was a real consideration in 1980);
because it wouldn't be possible to tell the difference between the two shapes.
the first reason is no longer relevant, of course, but the second is.
go ahead and define your own:
for latex, or
if you're using plain tex.