You can use the names of the nodes (C) and (S)` and then connect them

If you desire to make connections from within the individual picture elements it is better to use a pic instead of a usebox.
Below, I define two pics:
my square which draws three squares named -s1, -s2 and -s3, and
my circle which draws three circles named -c1 and -c2.
If you name each of the nodes with the indivudla pics you can connect them:

Code:
\documentclass[border=2mm]{standalone}
\usepackage{tikz}
\newsavebox{\cR}
\savebox{\cR}{\hspace{0.1ex}\tikz[baseline=0.1em]{%
\node[draw, circle, inner sep = 0, minimum size = 1.4mm](c1){};}%
\hspace{0.1ex}}
\newsavebox{\sQ}
\savebox{\sQ}{\hspace{0.1ex}\tikz[baseline=0.1em]{%
\node [shape=rectangle, draw, fill=gray!30, inner sep=0pt, minimum width=0.9em, minimum height=0.9em] (s1) {};}%
\hspace{0.1ex}}
\begin{document}
\begin{tikzpicture}
\node[inner sep=0pt, outer sep=0pt] (C) at (0,0) {\usebox{\cR}};
\node[inner sep=0pt, outer sep=0pt] (S) at (0,1) {\usebox{\sQ}};
\draw [red] (C) -- (S);
\end{tikzpicture}
\end{document}
Code: Using pics
\documentclass{article}
\usepackage{tikz}
\tikzset{Circle Style/.style={
draw,
circle,
inner sep=0pt,
minimum size=1.4mm,
}}
\tikzset{Square Style/.style={
shape=rectangle,
draw,
fill=gray!30,
inner sep=0pt,
minimum width=0.9em,
minimum height=0.9em,
}}
\tikzset{
my square/.pic={% Define a .pic to draw three squares and name each of them
code={
\node at (0,0) [Square Style] (-s1) {$s_1$};
\node at (1,0) [Square Style] (-s2) {$s_2$};
\node at (2,0) [Square Style] (-s3) {$s_3$};
}
},
my circle/.pic={% Define a .pic to draw two circles and name each of them
code={
\node [Circle Style] (-c1) at (0,0) {$c_1$};
\node [Circle Style] (-c2) at (1,0) {$c_2$};
}
},
}
\begin{document}
\begin{tikzpicture}
%% These two lines replace the \usebox
\pic (MySq) at (0,0) {my square};
\pic (MyCr) at (0,-2) {my circle};
%% Now you can connect between the pics ans even within a pic
\draw [red, thick, ->] (MySq-s1) -- (MyCr-c2);
\draw [blue, thick, ->] (MySq-s3) -- (MyCr-c1);
\draw [magenta, thick, ->] (MySq-s2.north) to[out=90, in=90] (MySq-s3.north);
\end{tikzpicture}
\end{document}
picis better. – egreg Mar 05 '20 at 11:43pic-based solution, as well. As far as I found out, apicis called as a whole entity similar to a\savebox, e.g., here: https://tex.stackexchange.com/questions/195923/tikz-how-to-create-and-reuse-a-picture/195935. So, the question does hold even in the case ofpics, i.e., how to access inner objects of a pic outside its definition scope. – Mar 05 '20 at 11:51