Can a NCO make a Soldier use nair to remove facial hair?

There's a product called magic shave that's basically the same thing, it's a depilatory cream, specifically used by people who have razor bumps issues.

But this NCO is setting himself up for some serious blowback in multiple ways. This Soldier also needs to make sure he stops pushing back his appointments

To answer your question, no.
Additionally, I find it extremely sad that there is someone in our ranks that legitimately thinks that nair would be a viable option.

This NCO is not a medical professional. If said soldier is having derm problems then he needs to go to sick call to be seen. Let medical do their thing. Temporary shaving profile on record while they sort it out and NCO accept what medical says as they are the ones in charge at this point.

When the NCO decided to make the soldier do something for a medical condition he has no tried to practice as a medical provider and also could have caused harm to the soldier. I get it he is upset the soldier is not meeting the grooming standards and it sounds like the soldier is not doing their part to make sure if they can’t meet the standard due to a medical issue to see a medical provider.

In the end though the biggest blame and harm is going to come to that NCO in this situation.

Some one is setting himself/herself up for an IG complaint. The soldier can be told to shave, until he gets a profile, he is stuck with shaving. If he chooses to shave with peanut butter and a plastic spoon and it gets him to conform to regulations who cares. As a leader we cannot tell you to use product X over product Y. Said soldier may need to try different things until they get their profile.

Pseudofolliculitis barbae? It's a common problem in men of color. There are other treatments that don't involve Nair.

First, I can understand why this NCO is attempting to enforce standards.
Second, this same NCO needs to pull his head out of his fourth point of contact. Products like Nair have warnings on them about NOT using them on the facial areas for a reason.
Third, there are better ways of handling this. Many companies make razors and/or other products specific for those who have problems with pseudofolliculitis barbae.

Hopefully the Soldier can get with Medical and find out exactly what is going on and how best to treat it.

Nair? On the face? Yea, let's decide to have the Soldier burn the shit out of his face by using Nair. Have the Soldier try different brands/types of razors before going with that stupid ass suggestion.

Woe unto the Officer, SNCO, or NCO who by word or deed essentially tells a medical officer to "%$#& off."

Sounds like some hazing is going on here. In my day, if a troop was caught unshaven, then they were forced to dry shave and wear the razor taped to the pocket of their BDU top. That, I would consider hazing. This happened to me once, but I wasn't damaged either because that's the way the Army was in those days.

Short Answer: Sick call, he can receive a shaving profile and authorized use of an electric razor. He can also have all this appropriately put in his records.
Long Answer: NCOs can not order a Soldier to take or use medications, or products outside of a profile or while not serving in a medical capacity under a license physician i.e PA, Nurse, Doctor, Surgeon. Potentially using hair removal products before a PA or above can assess the severity of the bumps can cause scarring or in a field/deployed environment infection. So just go to sickcall do PT later in the day.