Yes, naturally the Breen are neutral. The few canon lines about them show: No-one really trusts them. Not the Feds - but they invite Breen to formal events (reference is a dialog between Picard and an Admiral about a formal event involving a shuttle race, stating 'the Ferengi already tried to bribe the Breen pilot'), not the Klingons (tried to conquer Breen long ago, never saw these ships again, never bothered with it again) and not the Romulans (have the saying to never turn your back on a Breen).
A Breen ship was also said to have attacked a freighter on the way to Cardassia, so the Breen were not on the best terms with the Cardassians before the war. During the war, the Breen were very demanding and used the bad position of the Dominion to their advantage. They certainly didn't become good friends. Breen also employ labor workers of several species/factions in their mining facilities - including Cardassians.
Basically, they are simply no-one's ally - and that makes them neutral.
On the other hand, there are also a few minor references in both canon and novels that Breen are on good terms with the Ferengi - a clearly neutral species. A Breen wrote the only Rule of Acqusition not written by a Ferengi. The Breen sold Ferenginar warp technology. The reference above about the shuttle race. And Quark mentions Breen and Nausicaans as hired guns he could get. Since he knows the price, it can be assumed other Ferengi hired Breen; and that Breen are mercenaries like Nausicaans. Breen selling Ferengi warp drive technology also says the Breen were way more advanced at that time, travelled outside their territory for business matters and must have seen it as more useful to trade with Ferenginar than destroy or conquer it.
So, the Breen are only on clearly good terms with a technologically inferior, neutral species of pacifist traders (Ferengi history has no armed conflicts at all) though they have the means to simply destroy/conquer them and are said to be a 'warrior species' with a clear interest in conquest. They are likely receptive to bribery, and work as mercenaries. Pretty obvious gain strategy: Conquer those that are potentially capable of resistance in case of war; play it nice with those that pose no threat and give you money.
Beeing with no one allied doesnt make them Neutral.
Right now the klingons are not allied with one of the "big ones" to. The Romulans arent, and the Cardassians are not really.
And all of them have and always had diplomatic representation wich gets invitet. That doesnt make them neutral.
Neutrals are merchants and merchenaries and any organizations wich not depend to one big empire. Like Ferengie, Nausicans or the Orion syndicate.
Breen could be an own fraction.... wich probably nobody would want to play, or part of a Dominion fraction. Not neutral.
In most cases Breen are shown as bad guys who doesnt want to have anything to do with anybody, but a NEUTRAL fraction part has to be somebody who has business with EVERYBODY.
The Klingons don't need to be allied with a 'big one' because they are a 'big one' themselves.
'Neutral' just means not taking sides. It doesn't mean you play it nice with everyone.
The Breen were shown as 'the bad guys' most of the time, but that also goes for Nausicaans and even more the Syndicate. Ferengi were originally created as 'enemy' species, too. It just went wrong because people failed to see what's so scary about capitalism. For a Gene Roddenberry, it was, but he forgot that the average capitalist is watching his vision. Klingons lost their image as 'scary brutes' for the same reason - fans liked their society and code of honor too much - hence Nausicaans were even more brutal and savage, yet had no honor or society people could start to respect and like.
Look at it this way: Breen want power. Breen don't like anyone, especially they don't like anyone being in touch with them. Whoever tries to pass their borders goes missing. Breen totally have the means and will to conquer other worlds. Maybe not a big alliance or empire like the Federation, Klingon or Romulan Empire, but there aer several smaller unions and independant worlds they could take. What stopped the Breen for so long?
Now, they suddenly make alliances with Ferengi, Orions and Nausicaans. All well known for piracy and trade. They become eyes, ears and most of all money makers for the Breen all over the quadrant. They do business with everyone, but never take sides for longer than they get paid for. Slowly, they claim one monopoly after another, until many major resources are under Breen control. And when things start to get ugly - the fights over rare resources get worse, maybe draw even more parties in the war between Feds and KDF - the Breen come out of hiding with a masterplan and the ability to strike full force against all weakened war parties; and the means to last a damn long time, quickly rebuild ships and intelligence from all enemies due to the 'friendly infiltration' their traders and pirates did.
It would suit all parties involved.
The Orion Syndicate would be divided at this point - unwanted non-Orions that got kicked out of the organisation for being the wrong species. They'd certainly have a grudge, against the KDF for going along with this policy, and against the Feds/their homeworld (as they already burned all bridges with them by joining the Synicate in the first place).
The Nausicaans are clearly aggressive and have the will to conquer, but not the means. They have no organized society to orchestrate an invasion of their own.
The Ferengi never had a real battle fleet. They have the structured society (though it probably got shaky ground after Rom became Nagus), the resources for weapons and technology, but not the military experience and likely also no reason to start a war. A lucrative offer from old friends could change that - war is good for business, after all.
The Breen self are the puppet masters - they don't bother with the infiltration and meaningless conficts between others. They don't get their hands dirty, stay at home, build up a massive invasion fleet, maybe laugh about the stupid factions not seeing how their plan comes together, and then do what Breen do best: surprisingly launch a big attack.