How to make a square with circles using tikz?

I'm not sure, which region of your image you like to fill by gray color. Like the following image? Please advice!

\documentclass{beamer}
\usepackage{tikz}

\begin{document}
\begin{frame}
    \begin{center}
    \begin{tikzpicture}
\draw[fill=gray](0,0) rectangle (4,4);
\draw[fill=white]
        (4,1)   arc (90:180:1)  --
        (1,0)   arc (0:90:1)    --
        (0,3)   arc (270:360:1) --
        (3,4)   arc (180:270:1) -- cycle
        ;
    \end{tikzpicture}
    \end{center}
\end{frame}
\end{document}

Or you like to have the following inverse coloring like this?

\documentclass{beamer}
\usepackage{tikz}

\begin{document}
\begin{frame}
    \begin{center}
    \begin{tikzpicture}
\draw   (0,0) rectangle (4,4);
\draw[fill=gray!30]
        (4,1)   arc (90:180:1)  --
        (1,0)   arc (0:90:1)    --
        (0,3)   arc (270:360:1) --
        (3,4)   arc (180:270:1) -- cycle
        ;
    \end{tikzpicture}
    \end{center}
\end{frame}
\end{document}

Two more options for such a figure. One uses a clipping path while the second draws the rectangle as a node which allows to use the auto clipping property of a path picture command.

\documentclass[border=2mm]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\draw[clip](0,0) rectangle (4,4);
\filldraw[fill=gray] (0,0) circle (1cm);
\filldraw[fill=gray] (0,4) circle (1cm);
\filldraw[fill=gray] (4,0) circle (1cm);
\filldraw[fill=gray] (4,4) circle (1cm);
\end{tikzpicture}
%
\begin{tikzpicture}
\node[draw, minimum size=4cm, fill=gray,
    path picture={\foreach \i in {north west, north east, south west, south east} 
        \filldraw[fill=white] (path picture bounding box.\i) circle (1cm);}]{};
\end{tikzpicture}
\end{document}

I understand that you just want the outline of this, so let it be straight but adjustable:

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

\begin{document}
    \begin{tikzpicture}
        
        \def\r{1.5} % Radius of corner circles
        \def\l{5}   % Square length
        
        \draw[orange,ultra thin] (0,0) grid (\l,\l);
        
        \draw[line width=3pt]
            (\l,\r)     arc (90:180:\r)  --
            (\r,0)      arc (0:90:\r)    --
            (0,\l-\r)   arc (270:360:\r) --
            (\l-\r,\l)  arc (180:270:\r) -- cycle
            ;
    \end{tikzpicture}
\end{document}

If you want to play with both \r and \l, you would draw some interesting pictures (especially if you select a radius that is greater than half the square length.

length=5 and radius=1.5:

length=5 and radius=3:

(orange grid is just to see dimensions).