Is it an EO complaint if your NCO is making you do extra duties, for example, covering down for shifts so that the rest can do mandatory PT?

No.

Is it 'unfair'? Absolutely. Is it something that might adversely affect morale? Sure thing. Does it feel like he's being singled out because they are on a profile and can't do mandatory PT (that's an assumption on my part)? If so, then yes - and he may actually be in a situation where that's accurate.

But aside from griping, is there ANY indication that he's being singled out because of his race, color, sex (to include gender identity), national origin, religion, or sexual orientation? If there is, then 'Joe' might have valid grounds for an EO complaint. Did 'Joe' say that's why he feels the NCO is having him do the extra shifts?

It's not an EO complaint if PFC Snuffy is told to work another shift and cover down on PFC Badluck's duties because he is laid up in the hospital. It's not even an EO complaint if PFC Snuffy is the one that is told to take his mask off to confirm that the chemical agent really is gone as well.

Both instances fall into the category of military necessity. It may appear that they are being singled out to the Soldier that has to be the one to cover down on the shifts or take the mask off, but unless there is something that points at the reason Snuffy was chosen was for the EO categories listed above (and not simply because Snuff happens to be in one of those categories), it's not an EO complaint.

Without knowing the limits on why your 'Joe' can't be involved with the rest of the unit doing PT, it's really impossible to comment further. For example, does your Soldier have a walking dead profile and can't approach within 50 feet of a PT formation before collapsing on the ground? If not, then there would be a strong argument for there being a DA-6 rotation of individuals that covered down on shift while the rest went to do PT.

It's also possible that it could run afoul of being in the ballpark of harassment if there is no valid military purpose tied to it (e.g., no requirement for anyone to be on shift), which would be a legitimate reason to speak to his Commander about the unfair treatment.

Again, both are hypotheticals, but the EO route is out.

If this seems a bit harsh towards 'Joe', it's because in my experience the majority of situations where a lower enlisted Soldier approaches someone about "is this possibly an EO complaint" when there is no hint of it being one, they are usually trying to get out of something they just don't like doing and are looking for a "club".

Again, is it unfair, something that could adversely affect morale, something that there could be a better way of accomplishing, etc.? Yes. But not EO unless there IS something that points to it being discriminatory and done because of his race, color, sex, national origin, religion, gender identity or sexual orientation.

As recommended by others here, seek guidance from the Equal Opportunity Advisor within your unit or any available EO representative within your chain of command to assess this situation objectively.

My first instincts is to say no. But, for clarification:

1) Without divulging/violating HIPAA, what activity or activities can your Soldier not do that is excluding them from conducting mandatory PRT with the rest of the company?
2) What shifts are needing to be covered while the rest of the Company is doing mandatory PRT?

Regardless of my instinct, whomever the EOA/EOL for your unit needs to be consulted to ensure if this is their lane or not.