I went off to the supermarket and came back to a deluge of e-mail on this subject. A sampling:

Jonah,
I’m an active duty Army JAG, and while that doesn’t make me any kind of ultimate authority, I suppose I have some experience dealing with the intersection of the law and the military.
The most salient point, I think, would be the concept of a lawful order, and that Soldiers are not required to follow orders from commanders that are contrary to law or regulation. Something clearly contravened by the Constitution would be an unlawful order, even when given by the highest possible commander the President.
Now, as a matter of military criminal justice, all orders are presumed lawful unless proven otherwise, and it would be on the Soldier in question to show that the order was unlawful, whatever it might be, if and when he were punished/prosecuted for refusing to carry out an order by the President (or any other commander). As a practical matter, you can imagine it would be pretty unliekly for a Soldier to go out on such a limb unless we were talking about something painfully obviously illegal.
As to the SCOTUS, they don’t have command authority, and so they cannot give Soldiers orders. In the scenario which you describe of a clash between the Pres and SCOTUS, assuming the President issued unlawful orders, some subordinate commander, like the Secretary of Defense, would have to issue orders to… do whatever, and assuming those were not contravened by the Constitution or a relevant statute, they could be carried out.
Best,
[Name withheld]

And:

Hi Jonah:
I think the question that you posed is very interesting, although I think that the answer is extremely clear cut; the Supreme Court is not, ever, under any circumstances, in the chain of command. I can’t concoct enough faulty Constitutional premises to get there (and I’m not suggesting that you were trying to). The president is the Commander in Chief, period. The military has a chain of command and they must obey the Commander. The maybe more interesting scenario is one wherein the president, having been impeached, refuses to step down. Under that scenario I believe that a new president would be sworn in following impeachment (let’s assume it is the former vice-president), and that new president would order the impeached president removed from office (bodily if need be) using either Federal law enforcement agents (this might require a declaratory judgment by the court that the new president was installed Constitutionally) or the military if needed, or both. It would get even more hairy if there was a split in the military with some officers loyal to the impeached president (who would be claiming that he is still president and therefore still Commander) and the new president. I feel confident that the military would scrutinize the impeachment proceedings to make sure that they were conducted according to the Constitution and, if they were, that they would then obey the commands of the new president. It gets even more interesting if there is some kind of national disaster where members of Congress have been killed or are missing and there become issues with determining how many votes are needed or it is unclear if the president was actually impeached or impeached properly. For example let’s say we suffer a nuclear attack and the president refuses to retaliate and whatever remains of Congress wants to impeach him in order to install a president that will retaliate. DC is in disarray and bedlam prevails. Wow! Someone should make a movie.
But the day that the American people allow the courts to directly order the removal of the president because they think he has gone so far out of constitutional bounds is the day that the republic ceases to exist.

And:

Jonah,
Your respondent declares:
“In our democracy, the military must always obey the commander-in-chief. Officers are free to resign, as you say, but while they remain on active duty they must obey the guy who won the electoral vote in the last election.”
Technically, this isn’t true; and displays a deplorable lack of understanding of the way a constitutional democracy operates. The oath of office for an officer varies only slightly by branch of the military; the current Army oath (http://www.history.army.mil/faq/oaths.htm) is:
“I, _____ (SSAN), having been appointed an officer in the Army of the United States, as indicated above in the grade of _____ do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign or domestic, that I will bear true faith and allegiance to the same; that I take this obligation freely, without any mental reservations or purpose of evasion; and that I will well and faithfully discharge the duties of the office upon which I am about to enter; So help me God.”
While enlisted personnel do swear to obey the orders of the President and the officers appointed above them, an officer does not. It was my understanding as and Air Force officer that this was why issues like war crimes and unlawful orders were important for any officer to pay attention to, as the defense of “I was just following orders” is pretty clearly moot.
In our constitutional democracy, the military is one of the guardians of the Republic, and it is my fervent hope that they never have to execute their duties to defend the Constitution against the Commander in Chief, but, at least as individuals, it would almost certainly be their duty to do so.

And:

I agree with the reader that says that Congress should do a quick impeachment/trial vote. But assuming they had but the now removed president refuses to go, whose job would it be to remove. A complication that we have that may not be parallel in Honduras is our military is specifically restricted from domestic law enforcement by the Posse Comitatus Act. Here I would look to the Federal Marshalls under the Justice Dept reporting to the new president (by succession). But Honduras may have the Army fulfill that role.