I, _____, do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; and that I will obey the orders of the President of the United States and the orders of the officers appointed over me, according to regulations and the Uniform Code of Military Justice. So help me God.

Uniform Code of Military Justice, sections as noted:

809. ART. 90. ASSAULTING OR WILLFULLY DISOBEYING SUPERIOR COMMISSIONED OFFICER.
Any person subject to this chapter who--
(1) strikes his superior commissioned officer or draws or lifts up any weapon or offers any violence against him while he is in the execution of his officer; or
(2) willfully disobeys a lawful command of his superior commissioned officer;
shall be punished, if the offense is committed in time of war, by death or such other punishment as a court-martial may direct, and if the offense is committed at any other time, by such punishment, other than death, as a court-martial may direct.
891. ART. 91. INSUBORDINATE CONDUCT TOWARD WARRANT OFFICER, NONCOMMISSIONED OFFICER, OR PETTY OFFICER
Any warrant officer or enlisted member who--
(1) strikes or assaults a warrant officer, noncommissioned officer, or petty officer, while that officer is in the execution of his office;
(2) willfully disobeys the lawful order of a warrant officer, noncommissioned officer, or petty officer; or
(3) treats with contempt or is disrespectful in language or deportment toward a warrant officer, noncommissioned officer, or petty officer while that officer is in the execution of his office;
shall be punished as a court-martial may direct.
892. ART. 92. FAILURE TO OBEY ORDER OR REGULATION
Any person subject to this chapter who--
(1) violates or fails to obey any lawful general order or regulation;
(2) having knowledge of any other lawful order issued by any member of the armed forces, which it is his duty to obey, fails to obey the order; or
(3) is derelict in the performance of his duties;
shall be punished as a court-martial may direct.

FindLaw.org, on lawful orders:

Of all of these requirements, "lawfulness" is the most difficult, and the most important, to figure out. An order is "lawful" when it does not conflict with the U.S. Constitution or its laws, other military or admiralty laws, or international treaties binding on the U.S. Also, a lawful order is one which doesn't interfere with an individual's constitutional rights. All orders are presumed to be lawful (except patently unlawful orders, such as deliberately killing innocent civilians), and any service member that refuses to obey an order may be properly court-martialed.

However, if an order is unlawful, the service member who receives the order has a duty first to seek clarification of the order and, if still unlawful, to disobey it. A commander who issues an unlawful order is essentially ordering his subordinate to commit a crime, and the subordinate who obeys the unlawful order is just as guilty of the crime as the commander. That's the reason why the defense that soldiers "were just following orders" when they committed war crimes almost always fails.

However, this can be an ambiguous area for service members because whether an order is lawful is ultimately a question of law to be decided by a judge well after the fact. Also, a service member doesn't disobey an order by merely indicating an intention to do so. Instead, an order is disobeyed when the subordinate fails to perform the action described in the order.

In short, no, the orders wouldn't be obeyed.