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

I still at 74 call men sir and women mam. If I am with family I call by first names or nick name. I don't think rank at home in the family is the same as rank on duty. For sure on duty , one knows that an officer is being addressed use the rank not nouns.

Absolutely you must correct him

Long ago, when I was a PO3 (E-4), I was ascending a long set of (outside) stairs at New Londong, CT with two E-3s (from my unit) ahead of me. Two midshipmen were descending, and the two E-3s (deep in conversation) failed to salute or even to respond when the midshipmen challenged their failure to salute (they were already 10 or 15 feet uphill. The only reason I did not salute was that I was the mail petty officer and was carrying mail in packages in both arms. I stopped and apologized to the midshipmen. When I got back to the barracks, I let the two E-3s know they had failed (1) situational awareness - they had been so deep in conversation they never even saw the two midshipmen AND (2) military courtesy.
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I gave them a choice of going on report or assuming extra barracks cleaning details. They chose the latter. Likely a wise choice since they were awaiting submarine school and going on report would likely have washed them out before starting.
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The point is that when a salute or other military courtesy is appropriate, it should be rendered. IN OR OUT OF UNIFORM. Obviously saluting is not necessarily appropriate when out of uniform, but respectfully addressing officers is ALWAYS appropriate.

I keep forgetting when I read these posts that this is the new kinder gentler army. Why this is even a question in here tells a lot. Sad to see the military in its current form lacking the discipline that made us the best.

Captain Railey, there is no way that you can let this stand, every one was taught and learned the standards and respect! Airborne! HN

Especially as a lieutenant

The obvious answer is to correct him -- firmly! In fact that answer is so obvious that my first thought was much like several other responses questioning your leadership.

But perhaps there is more to this question that isn't immediately obvious.

What is this "outside work" situation -- is it some sort of off-duty military activity like working out at the post gym or is it a purely non-military activity? How does everyone else address you in this situation? Where did the two of you first meet, would he have known you were an officer when you first met, and did you know he was a SPC when you first met? Let's say, for example, that the two of you first met playing in a pick-up basketball game on a community basketball court in the civilian subdivision where you both happen to live, both in civilian clothes, with no reason for either to know the other is in the Army... This would set a very different starting point and require a different approach to reach the same solution.

Or we could try a different starting point. Suppose that pick-up basketball game was on post, you're in PT gear out for a run when you come across a small group of soldiers playing basketball, you decide to join the game. So, the SPC still doesn't know you're an officer -- you're acting like just-one-of-the-guys and that's exactly how everyone is treating you. Same problem, but now it's clearly all your fault.

The bottom line, however, remains that the situation cannot be allowed to continue; the difference is in how YOU would go about correcting the situation.

"You can lead a horse to water, but you cannot make them drink... however from time to time you may have to near drown them to prove a point"

WEAK LEADERSHIP MAKES FOR A WEAK MILITARY. There is a Chain of Command for a reason. Fraternization WILL NOT BE Tolerated! BUT while I was at Ali Al Salem Kuwait, I seen Air Force Enlisted calling Officers by their first names. I seen it, and I heard it and it made my head about explode.In the Navy if I heard this, I’m putting my foot up the enlisted and the Officers asses. I’ve gotten into an argument with a full Bird Colonel before. The United States Navy gave me a job to do, and by God I’m doing my job to the best of my ability. FRATERNIZATION WILL NOT BE TOLERATED.

I find it interesting how many people advocate for yelling at another service member in public. It doesn't matter what your rank is and their rank is, that is unprofessional and makes us look bad to the American people. I'm not a touchy-feely guy, I'm just a very big advocate of not being a dick. It also undermines your Authority to unnecessarily be discourteous. You can absolutely be professional and teach some of the proper way without raising your voice. The best response is to keep talking to the kid and then professionally correct him at an appropriate time in the conversation. Also, if you want to talk about professionalism you need to have the professionalism to read a sentence. The first sentence says outside work, and too many people just come out and say I assume this is at work. You cannot enforce one standard will neglecting others.
This is actually a very good teacher will moment, as many as pointed out, because his unit might how that is an okay thing to do in their culture. The right thing for you to do is to make sure he knows that not all units are like that so that he is not lit up by sergeant major and Colonels they are on.