When an E-4 calls me "hey" or "hey man" knowing that I am an officer, should I bother to correct him?

If you are having a personal conversation, say you both love to fish and are talking about that, or at a you are golfing together, talking at a church group, etc., then no. If it is in any other kind of situation that means it "work" since you are both in the military and then it most certainly needs to be addressed and quickly.

I know this is a year and half old, but I'll throw my experience in as well.

When I was in the National Guard, one of my golfing buddies happened to the the Battalion CO. We had a relationship outside of the Guard, my wife was a small town surgeon and my BC was on the board of the hospital. At the time I was a SPC and he LTC. On the golf course it was first names only, sometimes last names, but mostly first names only. We were just two guys out golfing. On duty, he could call me whatever he wanted to call me, he was the BC. On the rare occasion our paths crossed on duty, I called him "Sir" or "Colonel" and nothing else. The two aspects of our lives never crossed, nor should they have. (on the golf course we didn't talk Army business and on-duty we didn't talk family business) Neither my CO, nor my 1SG, not even my gun crew, knew that I hung out with the BC on odd weekends. It wasn't any concern of theirs.

Second experience, when I was a civilian police officer, my shift SGT was also my best friend. Away from work, drinking buddies, goof offs, etc. At work, "Yes Sgt", "No Sgt", I was a patrolman and he was my supervisor.

I think it comes down to the maturity of the two people. Can they handle BOTH a friendship and a professional relationship. The onus usually falls to the lower ranking person in the relationship. Can they handle that their friend is in charge and when you are at work/on duty your friendship cannot interfere with the chain of command nor the mission. Many people who are on the lower rank side of a friendship cannot handle that.

Grow some fucking fuzz on your peaches and set the damn line. If it bothers you, tell him. Fucks sake "Hey CPL, I don't mind it off duty, but mind your customs and courtesies, I don't wanna be put in the spot to need to address it in uniform". It's a common colloquialism he probably doesn't even notice.

Stop him/her and explain that you are not his buddy

I give them a chance. I reply, "Want to say that again?" Don't let him walk away unless he fully understands he's way off target. Part of the problem or part of the solution. A choice we are each given.

Remind him or her to say "Hey, man...sir."

Sir, if you have to ask this question there are several failures within you chain of command! First, the lack of discipline would be evident, no lower enlisted soldier should address any NCO or Officer in such a manner!

Your failure to understand what’s wrong with setting the basic standards is evident.
The failure for this enlisted soldier to follow protocol is evident!
I could go on, but let’s make it simple. Either correct the bad behavior or accept the fact that your inability to correct this soldiers bad behavior is why it continues.

Politely instruct this E4 of proper protocol. Then tell him/her this is a one time only offer. We can start this encounter over with proper protocol and be done with it. But this is the one and only time it will be tolerated. My feeling is we are trying to create a culture of respect and not disdain. A serious tail chewing for this first encounter will not correct the issue any better and as an officer you will be regarded as a jerk. Second offense pull out the rabid Rottweiler. Just my humble opinion as a former officer.

Before you wrote your inquiry, you already knew the answer. If you didn't think it was a possible problem, you wouldn't have posted it here.

You can choose to deal with the behavior in several different ways, but you don't want to ignore it.

why would he be comfortable calling you "Hey man" out of work? You are not personal friends. Now that would be understandable if he were an old friend that joined the military after you did and ended up in the same vicinity. Explain to him what he fails to see, tactfully first. Make this a teaching and learning moment. Anything else afterward is subject to disrespectful behavior.