Given:

$ z_1 z_2 = 0$

$\Rightarrow |z_1 z_2| = |0|$

$\Rightarrow|z_1||z_2| = 0$

$\Rightarrow|z_1| = 0 \vee |z_2| = 0$

We know (Modulus is non-negative; Positive for non-zero numbers):

$|z| = 0 \iff z=0$

Thus:

$ z_1 z_2 = 0 \iff |z_1| = 0 \vee |z_2| = 0 \iff z_1 = 0 \vee z_2 = 0$

Another method would be to use polar coordinates:

$z_1 z_2 = r_1 e^{i \theta _1} r_2 e^{i \theta _2}$

$ 0 =r_1 r_2 e^{i (\theta _1 + \theta _2)}$

Since $e^{z} \gt 0$ $\forall z$:

$ 0 = r_1 r_2 \iff r_1 = 0 \vee r_2 = 0 \iff z_1 = 0 \vee z_2 = 0$

I am sure using rectangular would also work, but it if probably more work and algebra than necessary.