This question has bugged me a lot and I'm not a 100% satisfied by MBaz's answer (the question is about white noise, and the answer is about flat noise) so I would like to add a few elements of answer.
One thing that bothered me a lot is that if you clip noise to a certain threshold, you're going to increase the probability of samples equal to your threshold. To me, this intuitively resulted in a spectrum change, because it increases the frequency of this particular value.
However, the amplitude repartition of a random sequence, which is to say its distribution, is not related to its self-correlation thus its spectrum.
For example, one might think that if you bit-reduce (quantize) a signal to an extreme extent, let's say from 64-bits to 2-bit, you completely change its color. But that is not true : bit reducing a signal changes its distribution but not is correlation properties (ie spectrum). Here is an illustration of 64 bits gaussian white noise's spectrum vs 2-bits uniform noise spectrum (values can be -1, 0 or +1 with the same probability) :
Similarly, if you hard clip white noise, you only affect the distribution and not the correlation which means the spectrum keeps the same color :
You do change the variance of the signal though, which changes the average power of the spectrum.
All the previous statements where based on ideal white noise, ie with infinite bandwidth. But that doesn't exist, and when you generate white noise digitally and then feed these samples into a DAC, interpolation (low-pass-filtering) is applied to the signal to transpose it into continuous time. If your hard clipping occurs after this filtering, then MBaz's answer applies and you get exactly what he explained. Here is an example with an oversampling factor of 4, and matlab's interp() function :
One reason why all this is a bit difficult to visualize might has to do with Gambler's fallacy
Please correct me if I made wrong assumptions.