Madame Squirrel is an original character created for the 2008 musical Emmet Otter's Jug-Band Christmas. She's the leader of a group of squirrels who perform an acrobatic act in the Frogtown Hollow talent contest.
During the 2008 rehearsal process, the characters evolved to the point where Paul Williams wrote a song for them. Various changes were made for the 2009 show (now known as Jim Henson's Emmet Otter) based on rehearsals, one of which was the naming of the other squirrels: Skippy, Junior, Jiffy and Nutella. Each of the characters was developed mostly by their respective puppeteers; the character names even appear in the program.
For the 2009/2010 season, the role of Madame Squirrel was taken over by Jennifer Barnhart.
While she does not appear in the original special, a lady squirrel seen cheering loudly for the acrobatic squirrels has a close-up in the original special. It can be assumed that this is who Madame Squirrel is based on.