HOA Nut House

The neighborhood has filed a petition for removal of Board Members. Those same Board Members have said they need an "objective 3rd party" to study the efficacy of the petition before they will consider scheduling the required "Special Meeting." Do they have the authority to do that? They are shutting out the other 2 Board members from communication.
If they do not schedule the special meeting in time, what happens next? Are they, by default, removed from their positions?
If the other 2 board members schedule that special meeting anyway, are there repurcussions legally on them?
Knowing there was a movement to remove them, the 3 members have hired a new management company with no Board meeting, no resolution to have the authority to do so (until after the fact), no due diligence, etc.
Can the homeowner's serve them with a Cease and Desist Order so they do not incur additional legal problems the homeowners will have to live with - after they're gone?