I would say it would come down to the same criteria as any other political decision, military or not. We'd decide our consciences and probably vote them, then act as units. If the violations are so esoteric as to not raise the ire of multiple folks you get the pleasure of acting your conscience as an individual, that would usually involve resigning your commission, going AWOL or attempting to get out through legal means.
When units mutiny, even over things like poor food or maltreatment, historically they tend to vote about it internally, democratic principles remain at work. Look back at those same Greek's who were free to go to war or not depending on what they felt about the arguments presented for and against.
In a situation where a significant number of the military feel that moral and Constitutional authority has been lost, they would probably rally to political leadership who state the same thing, hopefully leadership who state sincerely an intention to not "take power" but simply remove from power those who would subvert Constitutional law. That would be in allegiance to their higher duty to support the Constitution in precedence to obeying unlawful (UnConstitutional by action or lack of authority) orders.
The dream scenario would be when a significant number of representatives of the people actively abandon government and states declare secession in response to blatant tyranny. That gives an alternative representative civil authority for the mutineers to join, keeps it as legal as mutiny against sitting authority is likely to be. But we played that game once before...
