For some reason they don't have the team game assist record in the books, possibly because the early records from the pre-Geno era on that stat might have been a bit fuzzy. Don't know.
From checking just the 2000s results, I would guess the record is 34, though it would be nice to be able to check the non-Tourney box scores for the 1994-95 team that averaged 22 a game and might very well have been capable of getting into the 30s in a high-scoring game, though neither Jen Rizzotti or any teammate reached double-digits in a game that year. The 34 mark was reached against Western Michigan in the 2003-04 opener when the Huskies had 40 FGs. Next best in the 33 against Georgetown on Jan. 02, 2001. UConn also got to 32 a few times, most recently in last year's opener against Charleston, and also against Dartmouth in a 2005 tourney game.
From tourney box scores, the teams of the 1990s besides the 1994-95 team did not come close to cracking 30 assists, and in one game the total was 4 (ouch). Some of the early 2000s teams were ferocious at piling up steals, which if they lead to a solo breakaway basket will eliminate a chance for an assist on a FG.