You might want a supercircle with $x^k+y^k=1$. A circle has $k=2$. If you increase $k$, the corners move out. You would have to increase $k$ with size to get the "sharper corners" you refer to.
Alternately you can make a square and put tangent circles at each corner. Say you want the square to have a side of $2$ with circles of radius $\frac 14$ in each corner. The centers of the circle are then at $\left(\pm(2-\frac 14),\pm(2-\frac 14)\right)=(\pm \frac 74, \pm \frac 74). $ The sides are just $x= \pm2, -\frac 74 \le y \le \frac 74$ and the corresponding segments in $y$. The arcs are $(x\pm \frac 74)^2+(y\pm \frac74)^2=(\frac 14)^2$