The birds themselves? Not a whole lot. Chickens don’t even get infected by the virus — varicella zoster, a member of the herpes family — that causes the rash. (They can however, get an infection with similar effects called fowl pox.)
The name, rather, likely comes from chickens’ association with weakness and wimpiness*.
Just like we might tease our chicken-livered friends for chickening out from an adventure that we’re brave enough to handle, the name chicken pox distinguishes the milder infection from the more serious and deadly smallpox and syphilis (a.k.a. the Great Pox).
While this origin seems to be the most widely agreed upon and used, there are two more explanations sometimes given in medical texts. One is that the name came about because the rash’s lesions can resemble chickpeas. The other is that the Old English word giccan (“to itch”) either got corrupted into “chicken” or mixed up with the similar-looking Old English cicen (“young fowl”).
And pox, if you’re wondering, is a variant spelling of pocks, the plural of the Old English pock, or “pustule.”
* As for why we associate chickens with weakness, I haven’t been able to find anything solid, but the Straight Dope boards provide some interesting and amusing discussion on it.