Gaians are, more or less, ecoterrorists and environmental extremists. They are considered criminals by Bretonia and thus have not been allowed to land on Bretonian bases, etc. But just because somebody is a criminal does not mean they are not Bretonian - rather, they are Bretonian criminals.
While the Gaians may draw members from other Houses and various walks of life, they still reside in a Bretonian system (for now - we'll see if the Gallic Royal Navy captures it or not) and would be subject to Bretonian laws and courts of justice if caught and convicted of crimes. If anything, outside members do not make the Gaians non-Bretonian; rather, the Gaians make those outside members Bretonian.
Nationality has nothing to do with culture or ethnicity. It is something only defined by law, and while Bretonia would condemn the Gaians for their methods and ideology, they would still consider them Bretonian. In the end, that's what defines the Gaians.
You might want to be wary of allowing your vision of what you want the Gaians to be cloud your contributions to this conversation about what the Gaians actually are. People that truly care about a faction do that often.