I owned a pre-safety Marlin 1894 (1974 manufacture) for over 20 years. Did a lot of shooting with that gun, did a lot of work on it, know it so well inside and out I could probably carve a working replica out of billet steel if given enough time.
I, personally, would not recommend it. I'm sure plenty of people will argue this point, but I never felt the 1894 had the refinement or finesse I wanted from it, even after throwing $100s at it in labor and parts during the time I owned it. The action, even after being worked on, wasn't as smooth as my basic off the shelf bolt guns or pump guns, and I never felt like it was 100% reliable. Again you'll get many people disputing this fact but that's my opinion handling not only my 1894 but several others.
I know the Winchester is even worse in this regard, an even more complicated gun that's not easy to service. If it comes down to the 1894 or the Winchester I'd take the 1894 any day of the week.
The Henry Big Boy I can't speak to but they have a good reputation and may be worth a look. But honestly, if I were set on a .44 long gun I'd look at the Ruger 77/44 bolt gun or even a single shot Encore build. In the 21st century I feel either of those two are a better choice.
It took me 20+ years to realize I wasn't a lever gun guy, although I like the history and appreciate the guns aesthetically. So take my recommendations with a grain of salt and don't let me rain on your parade. If you do buy an 1894 let me suggest the Wild West Guns trigger right out of the box. Best bang for buck mod out there.