What is the smallest positive integer $n$ such that all the roots of $z^4 - z^2 + 1 = 0$ are $n^{\text{th}}$ roots of unity?
Multiplying the equation $z^4 - z^2 + 1 = 0$ by $z^2 + 1$, we get $z^6 + 1 = 0$.  Multiplying this equation by $z^6 - 1 = 0$, we get $z^{12} - 1 = 0$.  Therefore, every root of $z^4 - z^2 + 1 = 0$ is a $12^{\text{th}}$ root of unity.

We can factor $z^{12} - 1 = 0$ as
\[(z^6 - 1)(z^6 + 1) = (z^6 - 1)(z^2 + 1)(z^4 - z^2 + 1) = 0.\]The $12^{\text{th}}$ roots of unity are $e^{0}$, $e^{2 \pi i/12}$, $e^{4 \pi i/12}$, $\dots$, $e^{22 \pi i/12}$.  We see that $e^{0}$, $e^{4 \pi i/12}$, $e^{8 \pi i/12}$, $e^{12 \pi i/12}$, $e^{16 \pi i/12}$, and $e^{20 \pi i/12}$ are the roots of $z^6 - 1 = 0$.  Also, $e^{6 \pi i/12} = e^{\pi i/2} = i$ and $e^{18 \pi i/12} = e^{3 \pi i/2} = -i$ are the roots of $z^2 + 1 = 0$.  Thus, the roots of
\[z^4 - z^2 + 1 = 0\]are the remaining four $12^{\text{th}}$ roots of unity, namely $e^{2 \pi i/12}$, $e^{10 \pi i/12}$, $e^{14 \pi i/12}$, and $e^{22 \pi i/12}$.  The complex number $e^{2 \pi i/12}$ is a primitive $12^{\text{th}}$ root of unity, so by definition, the smallest positive integer $n$ such that $(e^{2 \pi i/12})^n = 1$ is 12.  Therefore, the smallest possible value of $n$ is $\boxed{12}$.