To enforce black-coloured text you can add \color{black} to the definition of \textdef@ from amsmath.sty.
To change the colour of the braces you can add that to the definition of \upbracefill (for \underbrace) and \downbracefill (for \overbrace). But I would go for defining a command, say, \underbraceColor that takes an optional argument to change the color.
I also changed \everymath{\color{blue}} to \everymath\expandafter{\the\everymath\color{blue}} to keep possible previous contents of \everymath there:

\documentclass{article}
\usepackage{amsmath}
\usepackage{xcolor}
\everymath\expandafter{\the\everymath\color{blue}}
\makeatletter
% To enforce black text (taken from amstext.sty)
\def\textdef@#1#2#3{\hbox{{\color{black}%
\everymath{#1}%
\let\f@size#2\selectfont
#3}}}
% To make the braces black (taken from fontmath.sty)
\newcommand{\overbraceColor}[2][.]{%
\mathop{\vbox{\m@th\ialign{##\crcr\noalign{\kern3\p@}%
\downbracefillColor{#1}\crcr\noalign{\kern3\p@\nointerlineskip}%
$\hfil\displaystyle{#2}\hfil$\crcr}}}\limits}
\newcommand{\underbraceColor}[2][.]{%
\mathop{\vtop{\m@th\ialign{##\crcr
$\hfil\displaystyle{#2}\hfil$\crcr
\noalign{\kern3\p@\nointerlineskip}%
\upbracefillColor{#1}\crcr\noalign{\kern3\p@}}}}\limits}
\def\downbracefillColor#1{$\color{#1}\m@th \setbox\z@\hbox{$\braceld$}%
\braceld\leaders\vrule \@height\ht\z@ \@depth\z@\hfill\braceru
\bracelu\leaders\vrule \@height\ht\z@ \@depth\z@\hfill\bracerd$}
\def\upbracefillColor#1{$\color{#1}\m@th \setbox\z@\hbox{$\braceld$}%
\bracelu\leaders\vrule \@height\ht\z@ \@depth\z@\hfill\bracerd
\braceld\leaders\vrule \@height\ht\z@ \@depth\z@\hfill\braceru$}
\makeatother
\begin{document}
\pagenumbering{gobble}
Some inline math $C_{0}$.
\begin{align*}
y = {\underbraceColor[black]{
\left[ x^2 + r^2 \left(a^2+\frac{a^{0.3}}{b}\right) \right]
}_\text{growth rate}} \times
{\underbraceColor{
\vphantom{ \left(\frac{a^{0.3}}{b}\right) } C_0
}_\text{initial}}
\end{align*}
\end{document}
If you prefer to set the color of the braces only once you can use this instead:
% To make the braces black (taken from fontmath.sty)
\def\downbracefill{$\color{black}\m@th \setbox\z@\hbox{$\braceld$}%
\braceld\leaders\vrule \@height\ht\z@ \@depth\z@\hfill\braceru
\bracelu\leaders\vrule \@height\ht\z@ \@depth\z@\hfill\bracerd$}
\def\upbracefill{$\color{black}\m@th \setbox\z@\hbox{$\braceld$}%
\bracelu\leaders\vrule \@height\ht\z@ \@depth\z@\hfill\bracerd
\braceld\leaders\vrule \@height\ht\z@ \@depth\z@\hfill\braceru$}