Is this fact about matrices and linear systems true? Let $A$ be a $m$-by-$n$ matrix and $B=A^TA$. If the columns of $A$ are linearly independent, then $Bx=0$ has a unique solution. If is true, can you help me prove it? If is false, could you give a counterexample? Thanks.

Yes. If $Bx=0$, then $x^T B x = (Ax)^T (Ax) = \|Ax\|^2 = 0$. Since the columns of $A$ are linearly independent, $Ax=0$ iff $x=0$. Hence $x=0$ and the solution to $Bx=0$ is not just unique, it is zero.