Correct me if I'm wrong, but didn't Clinton's budget plan from 1993 pass without any Republican support? So why is Lieberman claiming Clinton worked with Republicans to balance the budget.

Wikipedia agrees with my memory:

Clinton signed the Omnibus Budget Reconciliation Act of 1993 in August 1993, which passed Congress without a Republican vote. It cut taxes for fifteen million low-income families, made tax cuts available to 90% of small businesses, and raised taxes on the wealthiest 1.2% of taxpayers. Additionally, through the implementation of spending restraints, it mandated the budget be balanced over a number of years.