This is my field, so I can offer some advice on the subject.
A range of wild birds can get it. Pheasants and quail are wild birds. Generally speaking though, upland game birds are not common carriers or hosts of it. Waterfowl are, especially Canadian Geese.
The risk with avian influenza lies with domestic poultry operations. Chickens or turkeys are jam packed into a barn in close proximity. That's why it spreads so quickly in these situations. Its devastating. It wipes out entire flocks quickly. That's why you see such wild swings with egg prices all the time. Wild birds are outside and generally not tightly packed like domesticated poultry.
That being said, I would advise using gloves while cleaning birds and washing your hands thoroughly afterwards (which is something I've done for 20 years already). Although cases in people are rare, they can happen. There has not been a single documented case of a person getting it from having contact with a wild bird though. The cases with people we know of originated from workers in domestic poultry operations or more recently at dairy farms, because dairy cattle now have it.
I compare it to field dressing a deer. The chances of getting CWD is remote, but I still wear gloves and wash my hands in that situation too.