But there is no unknown distribution. There is a latent parameter (the n that's uniformly sampled) whose distribution is known; so the distribution of colors for the next draw is well defined. "Depend fully on the unknown distribution" is a statement about the distribution conditioned on that latent parameter; even in that case, the answer isn't "don't know" but "whatever the ratio of remaining balls given that parameter, and the one drawn ball, is."
"Don't know" isn't a valid answer in this case; "Don't know" would be valid if we were not told that n has that uniform distribution.
Since (unconditionally) red and green are symmetric, and the first draw is 50-50, the 'more likely/less likely' has the same meaning whether you interpret it as 'more/less than 50-50' or 'more/less than before.' This is convenient; if not for this symmetry, the answer would depend on interpretation of the 'more/less likely' language and could be ambiguous. The way the problem is stated, it is not ambiguous because either reading leads to the same result.
Let me try again. In the first interpretation the red ball drawing signals an absence of the n=0 case for the distribution, so we have a very slightly biased shift towards the red side of the potential n's, now in [0-99] vs [0-100] before the draw, so the question becomes does probabilistic gain in red bias distribution outclass the potential increase in green bias (reds now having reduced by 1 in the particular chosen n we are dealing with)?
The second interpretation says that given any selected n, the probability of drawing green of the next ball from that urn can only increase or stay the same (in the all red balls urn case) compared to drawing from that urn before.
It is the absolute vs relative interpretation of what the 'more' refers to.
I'll grant that the inclusion of 'uniform' for n in the question suggests the absolute interpretation as the relative interpretation would be independent of the uniformity of n.
The question could have eliminated ambiguity by just asking for the probability of the next ball color, but given the context of a Google recruitment test they might be more interested in how candidates reason under uncertain conditions rather than their dry probability skills.
"Don't know" isn't a valid answer in this case; "Don't know" would be valid if we were not told that n has that uniform distribution.
Since (unconditionally) red and green are symmetric, and the first draw is 50-50, the 'more likely/less likely' has the same meaning whether you interpret it as 'more/less than 50-50' or 'more/less than before.' This is convenient; if not for this symmetry, the answer would depend on interpretation of the 'more/less likely' language and could be ambiguous. The way the problem is stated, it is not ambiguous because either reading leads to the same result.