Whatever the exact nature of the Breen, we probably have to assume that both Kira and Dukat would have it in their interests to keep that nature secret.
I mean, even if the Breen evaporated when uncorked, Kira would no doubt have reported this. She's enough of a Starfleet officer in spirit by the time of "Indiscretion" to mind such details, and to write complete reports. And Worf, while always prone to quote myths and legends, would have learned of the "volatile" nature of the Breen from those reports.
Now, what could bring Kira and Dukat together on something like that? On anything, really? The greater the secret, the more painful, the more worth keeping - the greater the temptation to use it against the opponent at some point.
It doesn't sound likely, then, that the Breen would evaporate, S:AaB style. Or that they would in fact be nothing but hollow suits. Those "secrets" would be too benign that Kira and Dukat would agree to form their own little omerta circle over the fact.
It might be that the Breen look more familiar than anybody suspects. Perhaps Kira and Dukat agree that they shouldn't endanger or embarrass the species or group that is hiding behind those suits, because both feel protective about that species or group - but that would be a wonderful way for Kira to blackmail Dukat and vice versa, and thus wouldn't ensure silence but would in fact goad revelation. It would be more natural for both to agree that the people or phenomena behind the suits are too frightening to be revealed to the world. And since neither Kira nor Dukat is known to fear much (apart from loss of face), the secret may be frightening indeed.
Timo Saloniemi