I've heard it said that some Chess960 positions have a significant advantage for White, usually involving fast attacks against undefended pawns. However, since zugzwang positions do exist, it's theoretically possible that Black could have an advantage in some opening setups. Which leads to the question: are there any where he does? If not, are there any that give White less of an advantage than he has in normal chess?
It happens that someone tried to evaluate all 960 positions with an engine. Several positions got a score of 0.00, but none of them offered any advantage for black.
Of course, those conclusions are only the evaluations of Stockfish at 40 plies depth. It is a strong hint, but no definite proof, that Black is fine in the huge majority of 960-chess starting positions, but not better in any of them.
This result matches with the intuition that in a symmetric, non-developed position, with all pawns on the 2d and 7th rank, there is always something useful to do if you are on move: in other words, there cannot be mutual zugzwang.
If you accept statistical data taken from games played between computers as evidence, then the answer may be yes.
See this link for the data.
At the time of writing, White scores below 50% in 155 of the 960 positions. A sample of the scores from the RNKBBRNQ starting position is shown below.
Note that the average ratings for both sides are equal, so distortions in expected result due to playing strength should be negligible.
The answer is probably NO, although there is no way to "prove" that (yet). Zugzwang positions exist, but they appear with many fewer pieces in the board or in some special situations that are definitely not forced from the starting position.
The "statistical evidence" suggested by some answers is not something to take seriously, as samples are small and respond only to how humans play, which is far from perfect (take games by two complete novices and you won't see any advantage for White in classical chess)
No starting position can have an advantage for black, but it may be that white may not be able to establish a positional advantage like in standard chess. Also, any attack that black can do can be executed by white also (or if not at least stopped by correct play).
As for zugzwang, just as Evargalo said, it just not possible.
I play 960 quite a bit (though it may affect my opening) and have not encountered any such position. Therefore, it probably is just a myth that such positions exist.