IMO...the Dana 44 is a better choice for the H3. I say this for several reasons...cost, weight, size, fitment, etc. You must remember the H3 is a full-time 4WD and this fact should weigh heavily on your axle decision. Whatever axle you choose...it will be active 100% of the time. Its easy to overlook this when you see guys cramming D60s under Jeeps and hard-core rock rigs. The Dana 60 is a huge hunk of iron, and you'll be lugging it around every time you drive anywhere. Its weight and parasitic loss is in a whole different class from any of the other components on the H3 chassis (a big deal for us 5-cyl guys). Its obviously stronger but, its also a huge piece of rotating steel..and you can't shut it off. Unless you are doing extremely hard-core wheeling, its over-kill for the H3. It wouldn't be such a big deal if we had a 2WD-mode.
The Dana 44 seems to "fit" the mid-size layout of the H3. It can be built-up (chro-mo shafts) for comparatively little cost and this will greatly strengthen it for 35"+ tires and some mid to extreme off-roading duty. It will never be a Dana 60 (especially a built D60)...but it is far more practical. A 73-87 GM half-ton Dana 44 fits perfectly, and it is the correct width (67") without any mods. It also shares the same wheel bolt-pattern as the H3, which is another plus. Parts are cheap and everywhere.
Two guys can load a D44 in a truck. You'll need a fork-lift for a D60. Its that big of a difference.