Is a standard normal r.v. the only r.v which square to a chi-square r.v with 1 df?

Question

I know that if $Z \sim \mathrm{N}(0,1)$, then $Z^2 \sim \chi^2_1$.

1. Is the converse true: if $X\sim \chi^2_1$ is the square of a random variable $U$, then $U$ is necessarily standard normal? I think it is false, but it seems my professor thinks it is true. For example $U = |Z|$ is not standard normal, but $U^2 = |Z|^2 = Z^2= X$, seems (to me) a chi-square random variable. What is the truth?

2. Are $|Z|$ and $Z$ the only (real) random variables which square to $Z^2$?

Answer 1

Your counterexample seems correct to me. You can even do $U=ZW$ where $Z \sim N(0,1)$ and $W$ is some random variable that takes values $1$ and $-1$.