You sure it is not male on the Air Injection Iron Eater Filter valve? Male connection on the valve side is what is shown in the manual you posted.
And I see now they want the ferrule compression fitting to attach directly to a 3/8" male threaded tubing connection on the Iron Out and not go to 3/8" NPT. Not very common in 3/8" (much more common in 1/4" as that is the setup for an ice maker).
If you verify the connection is male on the Iron Out valve, you'll want to get a ferrule tube to insert into the end of a plain end 3/8" tube, a 3/8" female ferrule nut, and a 3/8" ferrule sleeve. Harder to find in small qty's on Amazon, but here is the smallest quantity I can find of a complete kit.
https://www.amazon.com/LTWFITTING-Compression-Insert-Sleeve-Ferrule/dp/B07FD18R9S