If $A^2$, and $B^2$ are DEPENDENT random variables, will $A$, and $B$ be necessarily DEPENDENT too?

I know that if $A$ , and $B$ are independent, the independence is preserved for $A^c$ , and $B^c$ , where $c$ is a constant. I am wondering if the same applies to the case where the random variables are dependent. I have been trying to check the relationship between two random variables $A$ , and $B$ . Using MATLAB and generating millions of samples, I get $\text{cov}(A,B) = 0$ , which tells me that $A$ , and $B$ could be either "independent" or "dependent with non-linear relation". Then I decided to find $\text{cov}(A^2,B^2)$ , and it is not 0. I know that if $\text{cov}(A^2,B^2) \ne 0$ , then, $A^2$ , and $B^2$ are dependent, and I am guessing that the dependency is preserved for $\sqrt{A^2}$ , and $\sqrt{B^2}$ . Is my guess correct? I suppose taking the square root could not make them independent. If my guess is correct, would the interpretation that the square root function is transforming a linear dependency into a non-linear dependency be correct too? The rationale would be that $\text{cov}(A^2,B^2) \ne 0$ , induces a linear dependency, and $\text{cov}(A,B) = 0$ , induces a non-linear dependency.