The employee should correctly fill out form CT-1040 based on the amount they had withheld, whether negative or not. If for some reason they had a "negative" amount of state taxes withheld, that means that they might owe more tax for the year and should include the negative amount on the form.
The only time that you enter "0" when a number is actually not zero, is when the form explicitly states something along the lines of
If the amount is less than "0", then enter "0"
The form makes no such statement for the lines where you input your total withholding for the year. The employee should make sure that their W-2 shows the correct amount withheld, and use that number for the 1040 form.
For example, the employee has a total of -$300 "withheld" for 2018. They would simply put "-300" in column C of line 18a. Unless they made additional tax payments outside of withholding or have tax credits, the total tax owed will probably be greater than the amount withheld and lines 26-30 of the form will apply.
The format of the negative number should probably just use a dash "-". While the form does not specify for withholding numbers, it does say that for business losses a negative sign should be used (as opposed to parentheses or brackets).
Also, I would get with the organization/people that do your payroll, as a "negative" withholding amount should probably not be allowed.