My "Wildlife of the Mid-Atlantic States" mentions a sandhill crane sighting in 1992, the first in some twenty years. But they are very rare, which I think lends credence to their being suggested as one possible source of the legend. You mention cougars, which were driven completely out of New Jersey in 1921; prior to this they occupied a very small range mostly in the north of the state. I agree that pumas have a weird cry- I'm a wildlife rehabilitator, and have worked with many cougars over the years. They are quite capable of killing several large sheep and countless chickens in the course of a single night. May I also suggest as a source for the livestock predation the packs of wild dogs which still roam the Barrens to this day? They were first noted in the mid-1800's as probable descendants of the dogs brought by colonists and Hessian soldiers. They are somewhat of a threat to backwoods hikers- the park service doesn't talk about them either, but residents do, and they are blamed for a great deal of mischief in those woods. Traps are set for them by rangers, but they've learned to avoid them.
Several members of the heron family make their home in the southeast of NJ, and in the marshy areas of the Barrens. Blue and night herons are rather large, some standing up to 2 ft., and they have a peculiar gait and the same piercing shriek most herons have. My own personal opinion on the JD is that it's a "Just So" story, created to explain unusual sightings or cries from the pines, which are unsettling enough without any legends. An alternative theory states that the "devil" dayes back to the late 1700's, when a local family (the Leeds of the story- they actually existed, and live in Galloway Twnsp. to this day- a descendant is a local politician) had a son who was retarded and badly deformed- a "devil" in the parlance of those times, which held any abnormality to be of satanic origin. Interestingly, there is some babsis for this- I have seen records of a Daniel Leeds being cared for through his life by his brother as he was "feeble-minded".
I'm not sure about a genuine cryptid in the Barrens- despite its protected status, it's a rather damaged and marginal habitat, mostly home to marginal species which thrive in disturbed environments, such as raccoons, deer, possum, etc. Heavy logging and cranberry-bogging in the past two centuries has driven out many species which once lived abundantly there, such as black bears, and in the earlier days, wolves. But if there were an undiscovered species in the pines, I'd have to guess it would be a bird, probably one of the herons or the wading birds.