Brownian noise is integration of gaussian or uniform white noise?
White noise is noise that has equal (uniform) amplitude across all frequencies. When we say "white" we're talking about the power spectral density (PSD) of the noise. 

Saying something like "Gaussian noise" means the statistical properties of any one sample of the noise is distributed Gaussian. You can actually have Cauchy, Poisson, Gaussian etc. distributions that define any one sample of the noise. For any distribution, however, if the power spectral density of the noise is uniform, that noise is white.

So in the integral defining the Weiner process:
$$W(t) = \int_0^t\frac{dW(\tau)}{d\tau}{d\tau}$$

$W(t)$ has samples with Cauchy, Poisson, Gaussian, etc. distributions as long as the power spectral density is uniform, making it white noise.