One wonders why.
The Office of Legislative Research issued a report late last month on, well, here's how they put it: If state law allows or ever allowed people to keep skunks as pets and any state allows people to keep skunks as pets.
The answer: No.
According to the report, prepared in response to a request from Rep. Linda Schofield (D-Simsbury), "Connecticut law prohibits people from keeping any wild quadruped without a Department of Environmental Protection (DEP) permit. DEP does not issue permits that allow people to keep skunks as pets, according to Chris Vann of the DEP.
Vann told the OLR that you can't even buy a skunk from a breeder in Connecticut, and that a law, CGS § 26-40, was passed in 1980 because rabid skunks were apparently "found in breeding facilities and retail establishments in the 1970s."
And though Connecticut is not alone, 17 states do allow skunks to be kept as pets, including New Jersey. 
There's even a skunk-owners advocacy organization, the American Domestic Skunk Association, Inc., which, as the OLR puts it, is "dedicated to assisting skunks and their owners."
Schofield said she requested the information because of a constituent's inquiry.
"There was a kid in my town who's allergic to cats and dogs, and wanted to have a pet" she said. "They wanted a cuddly, furry pet."
Rabbits, she said, aren't often that cuddly — "they have kind of sharp claws," she said —and the child had read about skunks and was interested.
"I had to ask OLR, why was it against the law," Schofield said. 
According to the Wikipedia page on the subject, they don't make bad pets.
"Skunks are sensitive, intelligent animals," according to the entry, taken from SkunksAsPets.com. "Skunks also tend to be very friendly, loving, entertaining and playful. However, they can also be stubborn and headstrong."
Despite this, the OLR report, written by Senior Legislative Attorney Janet L. Kaminski Leduc, stipulates that "skunks are one of the most common sources of wildlife problems experienced by Connecticut homeowners."
There is, as yet, no known vaccination against rabies for skunks. 
"They were disappointed," Schofield said of her constituents' reaction to the revelation that no, they could not adopt a skunk. "I basically explained to them that, even if they tried very very hard, even if I tried very very hard, the law was unlikely to change."