In my opinion, this is a very difficult question, since the piece values in antichess are not as static as in normal chess and usually highly depend on the game phase and position.
However, there are some rules of thumb which pieces tend to be stronger or weaker, see, e.g., http://poincare.matf.bg.ac.rs/~andrew/suicide/StanGold/theory.htm.
You can also try to derive the piece values from the results that are obtained in automated parameter tuning for antichess engines. E.g., there is a paper on automated learning of piece values for chess variants, also including antichess/suicide, which you can find at http://www.ke.tu-darmstadt.de/publications/reports/tud-ke-2008-07.pdf. They got only negative piece values for antichess, so their engine probably was not strong enough to find that it can be good to have more pieces. Due to the simplified evaluation function they used (with basically only piece values), the result is not unexpected, I think.
In 2016, I added antichess to the Stockfish fork with support for multiple chess variants that is also used on lichess. Over time we improved the search and the evaluation function and repeatedly ran an automated optimization (with SPSA) of the piece values. When the evaluation function was very simple and the playing strength rather low, the optimal piece values were all negative. As the search and evaluation got more sophisticated and playing strength improved, the optimal piece values changed and some moved towards zero or even to positve values.
Today, it seems to be the strongest antichess chess, and the piece values mostly agree with intuition of strong human players. You can find the current piece values in the code. Since there are several other evaluation terms (piece-square table, mobility bonus, etc.) that are influenced by the presence of pieces, these values are not completely representative, though.
Here is a summary of my interpretation of Stockfish's piece values (also trying to consider the influence from other evaluation terms):
- pawns are usually bad, since they are very restricted in their movements and are easy targets for forcing captures. I would give them a value of -1.
- knights are quite dangerous, but for a strong player they can also be very useful pieces. According to Stockfish, the knight's value is about -1 in the middlegame, and +2 in the endgame.
- bishops are usually very dangerous in the opening/middlegame, since the opponent can often get rid of all pieces if you have to start taking pieces with your bishop. Their value is about -3 in the middlegame and +2 in the endgame.
- rooks tend to cause the same issues as bishops in the middlegame, but are usually very strong in the endgame. Their value is -5 in the middlegame and +2 in the endgame.
- the queen can be useful in the opening due to her high mobility and the ability to quickly attack and capture pieces that are covered by the opponent, which might force him to take back in an unfavorable way. In the endgame the queen attacks too many squares and is usually bad. The piece value estimate is 0 in the middlegame and -2 in the endgame.
- the king is a strong piece because it can move to many squares, but only locally. In this way, it can be used to avoid zugzwang and to put up local threats without attacking squares in the opponent's camp. The piece value is something like +2 in the middlegame and +3 in the endgame.
You might find that the results disagree with some the rules of thumb I linked above. That only confirms that it is hard to break down the evaluation of pieces in antichess positions to a few numbers.