I think you are mistaken. University of Conn does need a writing sample. It even says so on their webpage. See below:
http://www.law.uconn.edu/admissions-/admission-scholarships/admissions-criteria-including-character-and-fitness

"In selecting the entering class, the Admissions Committee makes difficult choices among a large number of qualified applicants. These choices are made by balancing a number of factors, including: the LSAT score; the type, breadth, and depth of college or graduate courses taken and the grades received; academic honors and awards; writing ability; letters of recommendation from persons who know the applicant well (academic letters of recommendation are the most helpful for fairly recent graduates); work record, including military service, Peace Corps, and VISTA; college and community activities; and character and motivation. Although performance on the LSAT and in college or graduate school is important, selection is made after a careful review of the entire admissions file."