Refinancing with negative equity is possible, but it will cost so much it is not typically advisable. Negative equity simply means you owe more on an outstanding balance on a loan than the asset is worth. For example, a negative equity mortgage means your home is not worth enough money to cover the mortgage balance. This hinders the change of a successful refinance for a number of reasons.
You will be using the asset, whether it is a home, car or other item, to secure the new loan. The asset is likely not worth as much money as it was when you took the initial loan. This means the new loan will have to have a larger loan-to-value ratio than the old loan. For example, your initial mortgage may have been $300,000 on a $350,000 home. The home is now worth $250,000, and you still have $275,000 left on the loan. You will be seeking a $275,000 loan for a home worth less than that.
When you seek a loan with insufficient collateral, the rates are likely to be higher, not lower, than the previous loan. This means you will possibly be refinancing to a higher interest rate. You can do this if it is the only way to avoid default, but it will end up placing you in an even more compromised financial situation.