$ \bigcup_{n=1}^{\infty}\{z\in\Bbb C:z^n=1\}=\{z\in\Bbb C:|z|=1\}$?

Those two sets are certainly not equal.

$\displaystyle\bigcup_{n=1}^{\infty}\{z\,|\,z^n=1,n\in \mathbb N\}$ is countable as it is a countable union of finite sets, while $\{z/|z|=1\}$ has the cardinality of the continuum.

Your argument is correct.

For example, $e^{\sqrt{2} \pi i}$ is not in the first set.