If your noise has independent and identically distributed samples from a zero-mean distribution (for example Gaussian), it is white. Other definitions of white noise also require the distribution to be symmetrical, but that is not required for the spectrum to be flat. Clipping the samples of such white noise will only change the common probability distribution of the samples:
This will affect the mean of the distribution, but if both sides are clipped in a way that preserves the zero mean of the distribution, then there is no "coloring" of the spectrum by a 0 Hz peak and the noise remains white.
Example in Octave (MATLAB clone):
Create random variables from a Gaussian distribution:
N = 65536;
x = normrnd(zeros(N, 1), ones(N, 1));
Clip from above:
y = min(x, 1);
Clip symmetrically from below:
z = max(y, -1);
Plot values of Gaussian random variables (blue), clipped above (green), clipped symmetrically (red). Horizontal axis = index, vertical axis = value:
plot(1:N, x, 1:N, y, 1:N, z)
Plot spectrum of Gaussian random variables (blue), clipped above (green), clipped symmetrically (red). Horizontal axis = frequency, vertical axis = magnitude:
loglog(1:N/2, abs(fft(x)(1:N/2)), 1:N/2, abs(fft(y)(1:N/2)), 1:N/2, abs(fft(z)(1:N/2)));
The sequence that was clipped from above has a spectral peak at 0 Hz. Otherwise, the spectral envelopes are flat. This is not very easy to see because of the increasing density of plot points in the high frequencies in the logarithmic scale plot.