Is it possible to have political opinions in the military without a conflict of interest?

Yes COL (Join to see) it is possible to have political opinions in the military without a conflict of interest especially for those service members in the reserve components.
I realize you are most likely focusing this question on election politics.
However every organization of any moderate to large size has "office politics" extant much of the time. The US military certainly has office politics in play at every major formation from battalion/USAF squadron/etc. on up.
One thing that all service members need to be careful of is limiting political discussions with service members of your own rank and never trying to unduly influence those subordinate to you.
It is also very good to remember that all people are human and no candidate or office holder is subhuman and they all need to be treated with at least a modicum of respect.

Actually this points to a different perspective. The Pentagon, DHS, State, CIA, FBI, Justice, etc. have a bunch of incredible people who have been stifled for years. That's why morale is in the tank with a number of the good folk who depart and leave the ideologues behind who tend to be "yes men/women" who won't buck up to the boss. So the question becomes what do you fear more, the now or a possible loose cannon? All this tells me is the quitters should quit now because they already told me they're not good enough either way. The best people move the Country forward regardless. I want them in those jobs.

Military leaders nee to be pure intentioned and stoic in front of the troops and public.

"The words broadly echoed what flag officers have said in the past about the reality show star: “Personally, I hope no one will be called upon to serve under a President T… I can’t bring myself to type the words,” retired Rear Admiral John Hutson, who once served as the Navy’s top lawyer, told The Daily Beast in July.

To be sure, those views are not uniform. Commanders deployed outside the Pentagon said they hear enlisted troops enthusiastically support Trump. Some describe enlisted service members fighting with family or other soldiers in defense of the real talk from the real estate mogul. But the Pentagon is an unusual military posting, one where it is easier to spot a general than a corporal. And if the divide between the enlisted and officers is true, the former—the base of Trump’s military support—are not a well represented population within the headquarters of the United States military.

Regardless, such fervor about political matters is a jarring thing to hear at first from those in uniform; they serve in a part of government that urges service members to drop any sense of identity or partisan politics. It is unusual to see someone in uniform even say whether they are Republican or Democrat, and if they do, often it is whispered like a secret; the final case of “Don’t Ask, Don’t Tell,” if you will."

Political opinions are okay... Making public that you'll act like a petulant little child if the candidate you don't like wins... not okay. Especially if your opinion can be interpreted as being representative of your branch service.

Read the article and find it difficult to believe that, "more often than not, they [currently serving officers in the Pentagon] proclaim they will leave." If there are officers currently serving that espouse those sentiments, please, don't wait for an election to drop your papers. Do your nation a favor and leave active duty NOW! If an officer truly felt that way, than the right move would be to resign--foregoing the benefit of a retirement pension. to retire is to remain beholden to the officer ethic.

I have served as a commissioned officer under Presidents Reagan, Bush-41, Clinton, Bush-43, and Obama. All were very different ideologically and politically. But, my service has never been about the politics of our civilian executives. I espouse George C. Marshall's view that commissioned officers should be apolitical--certainly in public. Here is a critical look at military leaders and politics. http://strategicstudiesinstitute.army.mil/pubs/parameters/Articles/09winter/corbett%20and%20davidson.pdf

A long time ago, in a much simpler time...A mentor told me that an officer should participate in the political process by voting-then commit himself totally to supporting the will of the people following an election...

...that was back when anyone running for office basically fell a little "right" or "left" of the same camp.

The first CIC I served under was Clinton-Not someone I ideologically agreed with 100%, but overall, someone I at least felt fulfilled the minimum role of the office. More importantly, as a young officer, I felt his Defense staff represented a more holistic cross section of experience. Bush seemed like someone from "my side" in many respects...but there were key members of his administration with whom I vehemently disagreed. By comparison, Mr. Obama seemed less committed to being "part" of the system, and his administration, surprisingly biased.

In short, I feel the current Government is run more by ideology than ideas...more by rhetoric than results.

I think it's impossible for a person with any personal integrity, or reasonable intelligence for that matter (and let's hope that's most of our leadership in the Services)...not to be a bit distressed when a candidate from either side comes into office on a wave of uninformed, unbalanced, and let's face it-potentially dangerous opinions.

When asked, by younger officers, how I think "politics" affects them...I often respond by stating that the compromise which allowed for a standing military in the first place, and which is represented in our Oath of Office, is the cornerstone of a system which should always prevent the sort of horrors both sides feel possible should the "opposition" win an election. Even in the midst of our own Civil War, it was the intrinsic nature of the command element of both sides that allowed the war to end...and a re-unification to occur. When the politicians, and the people had seen their speeches play out into bloodshed...when the nation had suffered so much loss, it was the integrity of men like Grant, Lee, Hancock and Longstreet that led us (in a remarkably short period of time) to be unified enough to confront not one, but two world wars.

At the end of the day, no law has power unless those with the power to enforce it, act.

Just about everyone I worked in my military career espoused their political beliefs in small circles. Years ago as a Captain with the 3ID in Germany, I was flying with then BG Eric Shinseki to/from Grafenwoehr to discuss gunnery options. This was in 1992 after Clinton was elected. He made a comment that at least the Senate balance still allowed the Republicans the ability to filibuster. Any officer doing it in public is not smart and that is extremely rare. But I think we have a responsibility as citizens and those who have the more at stake than most Americans to be politically knowledgeable and exercise our right to vote. I have always disagreed with those who say that military should not vote. I would bet there are just as many military officers that are frustrated with Obama and his policies as there are who would be frustrated with Trump should he get elected. I came in under Reagan and retired under Obama so I have seen the entire spectrum. We have to remember that the military's duty is to serve the Nation and protect the Constitution.

I think so.... Look at the state of the army now...it's like I don't even recognize it anymore...Low morale....I see NCO's walking around the PX out of uniform, and other NCO's not making the correction...I do feel this is a result of our current leadership trying to make the army a social experiment.... last month My wife and I were walking through the PX and saw a SPC male with makeup on....Not one NCO made the correction....I did walk up to him and told him as a retiree...you are a embarrassment to the uniform

Possible, absolutely. Vocal... not so much.

We're literally electing the person who can send us to our deaths. And there is the real possibility of electing incompetence. That's scary.

Right now we have the advantage of saying whatever we want about whatever candidate we want, and not a soul can do anything about it. Our "frontrunners" aren't elected and don't place us within the confines of the UCMJ. The other candidates (Sen Cruz, Sanders, and Fmr Gov Bush) have are less polarizing and don't raise the vitriol level as much as the leads from each party, who are essentially "fair game."

After the election, that "release valve" is going to be closed, and closed hard, and the military is not going to be happy with any of the choices, because frankly we're never happy. We maintain a level of dissatisfaction from mild to extreme. And when we can't vent it just gets worse.