Draw a node as a square with TikZ

Use the regular polygon shape from the shapes.geometric library. As Alenanno mentions in a comment, you can define your own style to reduce the verbosity.

I don't know how the size of that is calculated, there is a lot of whitespace in it. Setting inner sep to a negative value can reduce the size, but the value has to be modified depending on the length of the node label.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\begin{document}
    \begin{tikzpicture}[square/.style={regular polygon,regular polygon sides=4}]
        \node at (0,0) [circle,draw] (c100) {$c_{100}$};
        \node at (3,0) [square,draw] (v100) {$v_{100}$};
        \node at (3,-5) [square,inner sep=-1.3em,draw] {a much longer node label};
        \node at (3,-5) [circle,draw] {a much longer node label};
        \draw (c100) -- (v100);
    \end{tikzpicture}
\end{document}

This is surprisingly difficult. In the following code, I've defined a custom node shape square that inherits from the rectangle shape, but forces its size to be square. Unfortunately, the inheritance mechanism doesn't allow redefinition of inherited saved anchors, so I had to copy some code from the TikZ rectangle definition.

On the plus side, it should be correct in all cases (defining different values for inner [xy]sep, minimum {width,height} and outer [xy]sep) and has all the anchors of the rectangle shape.

\documentclass[tikz,margin=5pt]{standalone}
\usepackage{tikz}

\makeatletter
% the contents of \squarecorner were mostly stolen from pgfmoduleshapes.code.tex
\def\squarecorner#1{
    % Calculate x
    %
    % First, is width < minimum width?
    \pgf@x=\the\wd\pgfnodeparttextbox%
    \pgfmathsetlength\pgf@xc{\pgfkeysvalueof{/pgf/inner xsep}}%
    \advance\pgf@x by 2\pgf@xc%
    \pgfmathsetlength\pgf@xb{\pgfkeysvalueof{/pgf/minimum width}}%
    \ifdim\pgf@x<\pgf@xb%
        % yes, too small. Enlarge...
        \pgf@x=\pgf@xb%
    \fi%
    % Calculate y
    %
    % First, is height+depth < minimum height?
    \pgf@y=\ht\pgfnodeparttextbox%
    \advance\pgf@y by\dp\pgfnodeparttextbox%
    \pgfmathsetlength\pgf@yc{\pgfkeysvalueof{/pgf/inner ysep}}%
    \advance\pgf@y by 2\pgf@yc%
    \pgfmathsetlength\pgf@yb{\pgfkeysvalueof{/pgf/minimum height}}%
    \ifdim\pgf@y<\pgf@yb%
        % yes, too small. Enlarge...
        \pgf@y=\pgf@yb%
    \fi%
    %
    % this \ifdim is the actual part that makes the node dimensions square.
    \ifdim\pgf@x<\pgf@y%
        \pgf@x=\pgf@y%
    \else
        \pgf@y=\pgf@x%
    \fi
    %
    % Now, calculate right border: .5\wd\pgfnodeparttextbox + .5 \pgf@x + #1outer sep
    \pgf@x=#1.5\pgf@x%
    \advance\pgf@x by.5\wd\pgfnodeparttextbox%
    \pgfmathsetlength\pgf@xa{\pgfkeysvalueof{/pgf/outer xsep}}%
    \advance\pgf@x by#1\pgf@xa%
    % Now, calculate upper border: .5\ht-.5\dp + .5 \pgf@y + #1outer sep
    \pgf@y=#1.5\pgf@y%
    \advance\pgf@y by-.5\dp\pgfnodeparttextbox%
    \advance\pgf@y by.5\ht\pgfnodeparttextbox%
    \pgfmathsetlength\pgf@ya{\pgfkeysvalueof{/pgf/outer ysep}}%
    \advance\pgf@y by#1\pgf@ya%
}
\makeatother

\pgfdeclareshape{square}{
    \savedanchor\northeast{\squarecorner{}}
    \savedanchor\southwest{\squarecorner{-}}

    \foreach \x in {east,west} \foreach \y in {north,mid,base,south} {
        \inheritanchor[from=rectangle]{\y\space\x}
    }
    \foreach \x in {east,west,north,mid,base,south,center,text} {
        \inheritanchor[from=rectangle]{\x}
    }
    \inheritanchorborder[from=rectangle]
    \inheritbackgroundpath[from=rectangle]
}

\begin{document}
    \begin{tikzpicture}
        \node[draw,square] {square with rectangular node text};
    \end{tikzpicture}
\end{document}

Use minimum size= which changes both height and width which you can change singularly as well. Also, keep in mind that rectangle is the default shape, so no need to specify it.

Writing \node at (3,0) [minimum size=1cm,draw] (v100) {$v_{100}$}; you should get