Positioning punctuation to the bottom right of an equation

You can always put the dot in the tikzpicture, making it sticks out without changing the bounding box with overlay:

\documentclass{article}

\usepackage{tikz}

\newcommand{\dummybox}{
\begin{tikzpicture}[baseline={([yshift=-0.45ex]current bounding box.center)}]
\draw (0,0) node[
    fill=white,
    rectangle,
    draw,
    minimum height=0.5in,
    minimum width=1in] {};
\end{tikzpicture}
}

\newcommand{\dummyboxwithfullstop}{
\begin{tikzpicture}[baseline={([yshift=-0.45ex]current bounding box.center)}]
\draw (0,0) node[
    fill=white,
    rectangle,
    draw,
    minimum height=0.5in,
    minimum width=1in] {};
    \path[overlay] (current bounding box.south east) node{\qquad .};
\end{tikzpicture}
}

\begin{document}

\begin{equation}
  \dummybox
  \quad = \quad
  \dummyboxwithfullstop
\end{equation}

\end{document}

Personally, I find putting the period in line with the equation number to be more apparent than at the bottom. Also, I make every effort to not put punctuation in equations, but sometimes it is unavoidable.

That being said, you can use a subscript.

\documentclass{article}

\usepackage{tikz}

\newcommand{\dummybox}{
\begin{tikzpicture}[baseline={([yshift=-0.45ex]current bounding box.center)}]
\draw (0,0) node[
    fill=white,
    rectangle,
    draw,
    minimum height=0.5in,
    minimum width=1in] {};
\end{tikzpicture}
}

\begin{document}

\begin{equation}
  \left.
  \dummybox
  \quad = \quad
  \dummybox
  \quad\right._{\displaystyle .}
\end{equation}

\end{document}