That's a new one on me so far as wearing baseball caps
backwards while in civies - Was it a military baseball hat?

The only AR that covers it as far as I know is 670-1
"c. Soldiers are associated and identified with the Army in and out of uniform, and when on or off duty. Therefore, when civilian clothing is worn, Soldiers will ensure that their dress and personal appearance are commensurate with the high standards traditionally associated with Army service. Commanders are charged with determining and publishing the local civilian clothing policy. When on a military installation, civilian headgear will be removed indoors in accordance with established norms"

Civilian clothing was once covered in garrison command policy. But, and it's been a while...I seem to remember a move away from that about 2010 and after a look at 4 different garrison command policy letter sets, I see none of them have a policy that speaks to civilian attire. To include Fort Eisenhower
AR 670-1 clearly states "Commanders are charged with determining and publishing the local civilian clothing policy. "
So again, check your local command policies Garrison to your company.

I had to double-check to see if this was a Duffleblog article or something similar.

Short answer: In almost all likelihood, there won't be. However, it CAN be prohibited by command leadership verbally or by published local regulations/policy.

There is no Service regulation that will govern civilian clothing in more than broad guidance (variations of "you represent the U.S. Armed Forces, don't embarrass us when you're in public wearing civilian clothes"). There may be local regulations or policy in effect, but that is going to vary between service components, units, locations, etc.

For instance, all personnel assigned to SOUTHCOM will adhere to SC regulation 670-1* while those assigned to Fort Jackson will follow Fort Jackson policy memorandum #24*. There is usually a limit on the geographical nature of those regulations/policy memorandums though.

To get to the perceived heart of your question though - Your unit or leadership can absolutely implement proscriptive or prescriptive restrictions on the wear of civilian clothing. However, this comes down to the overarching guidance on command policy and what leadership should and shouldn't concern themselves with. In the Army, much of that is covered in AR 600-20* and the other services have their own versions of that regulation.

However, even if there is no written policy/regulation, if command leadership has given a verbal command to do or not do something, then it is

Summarizing that and everything else that was beat into my head over 32 years, it comes down to a simple thing. The extent of your command's authority is extremely broad to the point of being almost unrestricted. Obviously it must be legal, ethical, and needs to have a military necessity, but those prohibitions are purposefully vague as commanders must have the latitude to act in the best interests of the mission, organization and mission. On the other hand, commanders are expected to exercise good judgement when it comes to exercising their authority and give weighty consideration to the impact of any order upon the unit and the morale of personnel, and should determine if "the juice is worth the squeeze".

So, while your unt/facility/installation/etc. leadership CAN give an order that baseball caps cannot be worn backwards while in civilian clothes, it is expected that the command leadership will give serious consideration to SHOULD they give that order.
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* SC regulation 670-1 - https://www.southcom.mil/Portals/7/Documents/USAG-M%20docs/SC%20Uniform%20Policy.pdf
* Fort Jackson Policy memo #24 - https://home.army.mil/jackson/application/files/6416/5114/9875/Policy_Memo_24.pdf
* https://www.armyresilience.army.mil/ard/images/pdf/Policy/600-20%20Army%20Command%20Policy.pdf