Are the birds in Minecraft chickens or ducks?

They are chickens. This can be verified in the source code, where the sounds they make are all identified as mob.chicken and the skin texture is chicken.png. Additionally, you can see their wattles in-game, which ducks do not have.

It has been mentioned that, in a recent blog post, Notch himself expressed some confusion over their species. As he is the one who created them, however, this is likely an in-joke for the community — he has a history of such humor, like the patch note for removing Herobrine.

Update: Notch tweeted today that "the chicken is now a duck". It's unclear whether this is a change in direction… or more trolling.

As pictured in the question, creature has a red piece of body unterneath beak. It's quite similar in placement and color to an organ named the wattle which distinguishes Chickens (Gallus Gallus Domesticus) from Ducks (Anatidae).

I think that they are chickens because in the minecraft wiki, it says that they are chicken.

Minecraft Wiki Mobs