$\newcommand\C{\mathbb C}$[Schwartz's Paley–Wiener theorem][1] states the following: 

>An entire function $F$ on $\C^n$ is the Fourier–Laplace transform of a distribution $v$ of compact support if and only if for all $z\in\C^n$ 
$$|F(z)| \le C (1 + |z|)^N e^{B|\Im z|} \tag{1}\label{1}$$ 
for some real constants $C>0,N,B$. The distribution $v$
in fact will be supported in the closed ball of center $0$ and radius $B$. 

In our case, $v:=u*u$ is compactly supported. So, by Schwartz's Paley–Wiener theorem, \eqref{1} with the Fourier–Laplace transform $\hat v=\hat u^2$ of $v$ in place of $F$ holds for some real constants $C>0,N,B$ and all $z\in\C^n$. Therefore and because $|\hat u|=\sqrt{|\hat v|}$, \eqref{1} with the Fourier–Laplace transform $\hat u$ of $u$ in place of $F$ holds with constants $\sqrt C,N/2,B/2$ in place of $C,N,B$ and all $z\in\C^n$. So, $u$ will be supported in the closed ball of center $0$ and radius $B/2$.  

  [1]: https://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem#Schwartz's_Paley%E2%80%93Wiener_theorem