$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}$No, there is no uniqueness here. 

Indeed, let $\hat f$ denote the Fourier transform of a (say integrable) function $f\colon\R\to\C$, so that $\hat f(t)=\int_\R e^{itx}f(x)\,dx$ for real $t$. Then $\widehat{u^*}(t)=\hat u(-t)^*$ for real $t$, and the equation 
$$u*u^*=A  \tag{1}\label{1}$$ 
for $u$ becomes the equation
$$\hat u(t) \hat u(-t)^*=\hat A(t) \tag{2}\label{2}$$
for $\hat u$. 

Clearly, equation \eqref{2} can have multiple solutions. So, equation \eqref{1} can have multiple solutions.