All US military personnel, as well as government officials (including the President) take an oath to support, defend and uphold the Constitution of the United States. Therefore, the Constitution trumps everything. Of course, the Constitution is silent on so many issues that there is rarely, if ever, a clear conflict between it and Presidential orders (which military personnel, as part of their oath, are sworn to obey).

The Constitution. Every soldier and public servant swear an oath: "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; that I take this obligation freely, without any mental reservation or purpose of evasion; and that I will well and faithfully discharge the duties of the office on which I am about to enter [so help me God]." A solider is DUTY-BOUND to refuse to follow any order that is inherently illegal or otherwise improper, wherever that order may originate from.

The Constitution. But defy a direct order from a superior that was "consistent with military objectives and national policy goals" and you've bought yourself a one-way ticket to a Court Martial with the verdict a foregone conclusion. You'll be making big rocks into little rocks for the next 20 years. It's something of a Catch-22: if ordered to commit an atrocity in pursuit of an operational objective (though what a government and nation that has carpet-bombed, nuked, and napomed civilian populations calls "an atrocity" is largely a matter of convenience or a sudden spasm of conscience), if you don't obey, you get Court Martialed and convicted for disobeying a direct order (at least) and if you do obey, you get scapegoated by the big boys if the public and press recoil in horror, and you're subsequently Court Martialed and convicted of manslaughter or mass murder. And in case you hadn't noticed, government and government officials for the last 20 years have shown an increasing disregard and even contempt for the Constitution that is only suprassed by their contempt and disdain for "the people". Of course the people themselves for the most part are either ignorant of or indifferent to the Constitution and Constitutional principles ... until petty tyrants hit them or their loved ones, or their crushed by the increasingly officious and despotic machinery of state. Just the fact that the Constitution requires a declaration of War by the Senate to do what we've been doing in Iraq and Afghanistan for the last several years proves that every US Serviceman who hasn't mutinied, deserted, or defied and been sent to Levinworth, either doesn't really know his duty, doesn't care, or is too afraid to do it. The problem is further complicated by the fact that the government AND the schools all teach that "the Constitution means what the government says it means", with the implication that it's just too arcane and sophisticated a document for the great unwashed to really understand it, and that we need a secular "priesthood of the law" to interpret it for us, and we should all just accept their pontifications unquestioningly and obey without hesitation or doubt. The only challenge may come from withing the ruling elite itself. So it is not for a serviceman, not even a General or Flag Officer, to interpret the Constitution for himself and do as his conscience dictates: "Defending the Constitution" now means "Implementing the will of those in power" -- as what they decree is ipso facto "Constitutional".

Addendum: Under historic military law, while enlistedmen have no choice but to obey an order (they're just cogs in a machine: only the officer who gave the order should be held responsible), a commissioned officer has had the option of "asking to be relieved of his command" rather than obey and implement an order from his superior. Up through WWII, this was still accepted and practiced. (In fact, 32 Admirals in sequence told FDR to got to hell when he ordered them to move the whole Pacific fleet to Pearl and put every capital ship into overhaul. FDR finally found a man dumb enough and who wanted the job bad enough to do it, Kimmel, who got only slightly what he deserved when he was fired a few months later after Dec. 7, making him the only 4-star scapegoat in US history.) Today, however, asking to be relieved will get you Court Martialed on multiple counts. Only Flag and General officers now have some measure of immunity, but they still risk demotion and loss of pension and benefits, and any other punishment the regime can possibly get away with inflicting on them. Instead, most just resign quietly as soon as they can.