*The answer is "no":* 
  
Take $M=6>5$ and evaluate
$$\mathbb{E}[|z_1|^2]=\mathbb{E}[|z_2|^2]=\frac{1}{60},$$
$$\mathbb{E}[|z_1|^2|z_2|^2]=\frac{1}{3360},$$
$$\Rightarrow\mathbb{E}[|z_1|^2|z_2|^2]-\mathbb{E}[|z_1|^2]\mathbb{E}[|z_2|^2]=\frac{1}{50400}\neq 0,$$
so $z_1$ and $z_2$ are not independent.