But the text is aligned; the boxes from the hyperlinks extend downwards, but the vertical position of the text remains unaltered, as can be seen with some auxiliary lines:
\documentclass{article}
\usepackage{tabulary}
\usepackage{booktabs}
\usepackage{hyperref}
\usepackage{tikz}
\begin{document}
\begin{tabulary}{6in}{rCJ}
\toprule
\textbf{Thing} & \textbf{Another} & \textbf{Thing} \\
\midrule
DEF & \hyperlink{def:ABC_DEF}{\texttt{ABC\_DEF}} & Some letters going on over here. \\
GHIJKL & \hyperlink{def:ABC_GHI}{\texttt{ABC\_GHI}} \hyperlink{def:ABC_JKL}{\texttt{ABC\_JKL}} & This box has enough text inside of it to do some word wrapping I should hope, as that's what happens in my table. \\
\bottomrule
\end{tabulary}
\begin{tikzpicture}[remember picture,overlay]
\draw[blue] (0,31.5pt) -- +(\textwidth,0);
\draw[blue] (0,19.3pt) -- +(\textwidth,0);
\draw[blue] (0,8pt) -- +(\textwidth,0);
\end{tikzpicture}
\end{document}

Perhaps you could suppress the hyperlink boxes using some of the hyperref options, as in, for example, \usepackage[colorlinks]{hyperref}:
\documentclass{article}
\usepackage{tabulary}
\usepackage{booktabs}
\usepackage[colorlinks,linkcolor=cyan]{hyperref}
\begin{document}
\begin{tabulary}{6in}{rCJ}
\toprule
\textbf{Thing} & \textbf{Another} & \textbf{Thing} \\
\midrule
DEF & \hyperlink{def:ABC_DEF}{\texttt{ABC\_DEF}} & Some letters going on over here. \\
GHIJKL & \hyperlink{def:ABC_GHI}{\texttt{ABC\_GHI}} \hyperlink{def:ABC_JKL}{\texttt{ABC\_JKL}} & This box has enough text inside of it to do some word wrapping I should hope, as that's what happens in my table. \\
\bottomrule
\end{tabulary}
\end{document}

If you want to keep the boxes, but making them tighter, you can use Heiko Oberdiek's suggestions in his answer to \ref with tight box surrounding the reference:
\documentclass{article}
\usepackage{tabulary}
\usepackage{booktabs}
\usepackage{hyperref}
\begin{document}
\hypersetup{pdflinkmargin=0pt}
\begin{tabulary}{6in}{rCJ}
\toprule
\textbf{Thing} & \textbf{Another} & \textbf{Thing} \\
\midrule
DEF & \mbox{\hyperlink{def:ABC_DEF}{\texttt{ABC\_DEF}}} & Some letters going on over here. \\
GHIJKL & \hyperlink{def:ABC_GHI}{\texttt{ABC\_GHI}} \mbox{\hyperlink{def:ABC_JKL}{\texttt{ABC\_JKL}}} & This box has enough text inside of it to do some word wrapping I should hope, as that's what happens in my table. \\
\bottomrule
\end{tabulary}
\end{document}
