Why are the real- & imaginary part of complex white gaussian noise independent processes?
It is simply because each sample of $n(t)$ has a random magnitude and phase by definition given as $n(t) = |n(t)|e^{j\phi(t)}$.  With real and imaginary components as follows:

$$|n(t)|e^{j\phi(t)} =|n(t)|\cos(phi(t))+j|n(t)|\sin(phi(t)) $$

Real: $I(t) = |n(t)|\cos(\phi(t))$ 
Imag: $Q(t) = |n(t)|\sin(\phi(t))$ 

Since the phase and magnitude are independent for each sample, the I and Q components will also be completely independent. Given any I value, there is no constraint or dependence on what the Q value can be for that sample, and vice versa. (A dependence would exist if given the phase, or given the magnitude. And given both then there would be a one-one mapping between I and Q)

Consider the opposite case if the I and Q components of a Gaussian Noise process were dependent such as I = kQ, the resulting noise would stay on a fixed angle passing through the origin (such as staying on the 45° line if I = Q) rather than adding a random magnitude and phase to each sample.