Square integral symbol

You can always build your own symbol. (If you want such integrals also to show up as subscripts, you need to add the corresponding code to the last options of \mathchoice.)

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}
\DeclareMathOperator*{\SquareInt}{\mathchoice{\tikz[baseline=0.55em]{%
\draw[line width=0.08em] (-0.2em,0em) 
|- (0,-0.2em) -- (0,1.9em) -| (0.2em,1.7em);}}{\tikz[baseline=0.35em]{%
\draw[line width=0.06em] (-0.2em,0em) 
|- (0,-0.2em) -- (0,1.2em) -| (0.2em,1em);}}{}{}}
\begin{document}
\[ \SquareInt\limits_a^bf(x)\,\mathrm{d}x\quad\text{vs.}\quad 
\int\limits_a^bf(x)\,\mathrm{d}x\]
$\SquareInt\limits_a^bf(x)\,\mathrm{d}x\quad\text{vs.}\quad 
\int\limits_a^bf(x)\,\mathrm{d}x$
\end{document}

You can design your own using \rules. This solution measures the height and depth of the integral to ensure exact size.

Here is the code:

\documentclass{article}

\usepackage{calc} % needed to add lengths 
\usepackage{amsmath} % needed for \DeclareMathOperator

\newlength{\hght}
\newlength{\dpth}

\DeclareMathOperator{\sqrint}{\mathchoice%
{\settoheight{\hght}{$\displaystyle{\int}$}\settodepth{\dpth}{$\displaystyle{\int}$}% measures \int
    \rule[-\dpth]{.2ex}{.5ex}% lower vert
    \rule[-\dpth]{.12em}{.2ex}% lower hor
    \rule[-\dpth]{.2ex}{\hght+\dpth}% main vert
    \rule[\hght-.2ex]{.12em}{.2ex}% upper hor
    \rule[\hght-.5ex]{.2ex}{.5ex}}% upper vert
{\settoheight{\hght}{$\int$}\settodepth{\dpth}{$\int$}%
    \rule[-\dpth]{.15ex}{.4ex}%
    \rule[-\dpth]{.1em}{.15ex}%
    \rule[-\dpth]{.15ex}{\hght+\dpth}%
    \rule[\hght-.15ex]{.1em}{.15ex}%
    \rule[\hght-.4ex]{.15ex}{.4ex}}%
{\settoheight{\hght}{$\scriptstyle{\int}$}\settodepth{\dpth}{$\scriptstyle{\int}$}%
    \rule[-\dpth]{.1ex}{.3ex}%
    \rule[-\dpth]{.08em}{.1ex}%
    \rule[-\dpth]{.1ex}{\hght+\dpth}%
    \rule[\hght-.1ex]{.08em}{.1ex}%
    \rule[\hght-.3ex]{.1ex}{.3ex}}%
{}}

\begin{document}

Inline square integral: $\int_0^1\sqrint_0^1$, subscripted: $A_{\int_0^1\sqrint_0^1}$, or a displayed version:
\[
\int_0^1\sqrint_0^1
\]

\end{document}

Note that I have not added the code for a scriptscript version. You can adjust the lengths and thicknesses as you wish.