Henne vs. Huhn
The German words Henne and Huhn both relate to the bird we call a chicken in English. However, they are not interchangeable, as they differ in their level of specificity, similar to the difference between woman and person.
Henne
A2Henne specifically refers to an adult female chicken, the one that lays eggs. It is the direct equivalent of the English word hen.
Die Henne legt jeden Tag ein Ei.
(The hen lays an egg every day.)
Im Hühnerstall gibt es einen Hahn und zehn Hennen.
(In the chicken coop, there is one rooster and ten hens.)
Eine Glucke ist eine Henne, die auf Eiern sitzt, um sie auszubrüten.
(A broody hen is a hen that sits on eggs to hatch them.)
Die alte Henne gackert laut im Garten.
(The old hen is clucking loudly in the garden.)
Huhn
A1Huhn is a more general term. It can refer to the species chicken, a single chicken of unknown or irrelevant gender, or the meat from a chicken. The plural is Hühner.
Zum Abendessen gibt es heute gebratenes Huhn.
(For dinner today, we are having roasted chicken.)
Pass auf, da läuft ein Huhn über die Straße!
(Watch out, a chicken is crossing the road!)
Der Bauer züchtet viele Hühner auf seinem Hof.
(The farmer raises many chickens on his farm.)
Das Huhn ist ein weit verbreiteter Vogel.
(The chicken is a very common bird.)
Mein Kind hat ein Bild von einem Huhn gemalt.
(My child drew a picture of a chicken.)
Summary
In short, use Henne when you are specifically talking about an adult female chicken (a hen). Use Huhn as a general term for any chicken, the species as a whole, or the meat. Every Henne is a Huhn, but not every Huhn is a Henne.