The basis for any sort of Japanese soup dish is going to be dashi stock, a stock made of fish and seaweed. It has a much lighter flavor than chicken broth, so you might not have identified it easily. You can buy it in a powder form for convenience, and it can be sprinkled into other liquids rather than reconstituted.
This recipe is for soba with a dipping sauce, but it'll give you a good idea of the flavor profile you're looking for: Dashi stock plus kaeshi, which is made of soy sauce, mirin, and sugar (and simmered so it's not strongly alcoholic).
It's down to what the chicken ate while it was alive. Saturated fat sets, olive oil sets if you chill it, but not otherwise, and a number of seed oils do not set (rapeseed for instance). When you make a stock which has solidified fat on top, that's saturated fat, so I'd hazard a guess that the stock where the fat doesn't set means a healthier eating chicken, because it contained less saturated fat to start with.
UPDATE: Thank you to the person who bothered to do the research and said my answer 'might have some merit'. Chickens are no different from human beings - the fats you put in are the fats floating round your bloodstream and depositing in various places; think about corn fed chickens, where the fat composition is slightly different, not to mention the colour of the flesh itself. That will be a partial explanation; when taken together with the fact that not all chickens, even in the same flock or brood, get to eat the same diet, because the pecking order dictates that some free range birds don't always get the pick of the food, explains differences in chickens from the same supplier. Of course, if you can come up with another explanation, I'd be delighted to hear it...
UPDATE 2: Perhaps I should have been clearer. I am not for a moment suggesting that the fats eaten are deposited in their original form, but if you know anything about biology (chickens or otherwise) then you'll know that certain synergies occur, depending what's put in, which change the composition of any fats deposited within the body system. Hence the connection between eating lots of saturated fat and having high cholesterol in humans, for example.
UPDATE 3: Rumtscho: Can't find any scientific evidence so far to prove this theory regarding chickens, but, for interest's sake, and to prove how much of a difference it can make, farmed salmon in Britain no longer has a balanced omega 3/6/9 ratio, as it should do, and still does in the wild. It's because the feed had to be changed, and the consequence of that has been a much higher level of omega 6 in particular. I'm still looking for something on chicken.
UPDATE 4: Now I've had time to look properly, it's not at all difficult to find scientific evidence, there's plenty of it. There's a study carried out by The American Society for Nutritional Science in 2000 comparing the fat deposition (and other metabolic processes) between chickens fed the same diet, but one lot with saturated fat included in the form of tallow, and the other lot with polyunsaturated fats. The fat deposition in the birds fed tallow was greater, and the composition of the fat contained more saturates compared to the polyunsaturated group. These results reflect previous studies (Sanz et al 1999 and 2000).
Effectively, it's like everything else - you get back what you put in.
Best Answer
See no reason why not, all you will do is add more flavour. Which is never a bad thing if you ask me. Adding the ham bone will not have any adverse effects either. The recipe you are following will not have used chicken stock/broth to either make it vegetarian or for simplicity.
Personally any time I make soup or broth at home I will use chicken stock as a base even if its just an oxo cube. At work ill use vegetable stock to keep those non-meat-eaters happy.