Certainly there is always the possibility that a random company here or there might go rouge. There have been examples of smaller groups going rouge even in the last 30 years. However, even that I would put on the highly unlikely scale. I don't consider those as a meaningful fracture. The general population probably wouldn't even be aware of these as these would be handled through internal diciplinary matters.

Considering the diverse geographic, ethnic makeup of all branches it is difficult to imagine that an entire batalion or brigade much less a division would choose to follow illegal orders. The military command structure is very different than it was in 1861.

The wild card, as always, is the National Guard but even if a few governors got some crazy ideas in their head it would be quickly put down and again it's doubtful that many guardsmen would follow illegal orders.