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.