Some years ago, one of our European CAS shooters was fortunate enough to get a tour of the Uberti factory (may have been pre Baretta), and observed about 4 - 5 Henry barrel units in 44 Special. He made a special request, which was honored, and some time later became the happy owner of a .44 Special Henry. Other than that "one" gun, Uberti hasn't made a Henry in .44 Special in a gazillion years. Every once in a while, one will come up for sale. You have to be very quick to score.
Currently, Uberti sells everything they make. They only make what they can sell. While there is reasonable demand for .45 Colt and .44 WCF rifles, the demand for .44 Special is ......... tiny. I placed a special order with Uberti some years ago for a .45 Henry Trapper. Took over two years to get. I wouldn't hold out hope of actually receiving a .44 Special Henry from Uberti before you achieve retirement age.
Like Abilene, I settled for a '66 in .44 Special. AS soon as they came available (previously made my own) I stuck a special carrier in my '66 to be able to run 44 Russian cases. I have 4 Henry's, all in .45 Colt, with special carriers installed so I can run Cowboy 45 Special cases.
Super fun to load on Sunday and shoot all week!
Coffinmaker
PS: OOPS..... the original question. The cleanest running, easiest to maintain would be 44-40. If one is already shooting 44-40 in other guns, adding a .44 Henry is a no-brainer. My choice however, is .45 Colt. Cause I'm immediately gonna modify it to run C45S cases. I only find the Blow-By of the straight wall cases to be a nuisance. It's why we clean guns ... right???