You can most likely use the same plug from your time clock to the new power unit. You just want to make sure of the voltage as most newer systems all come from the factory at 220 volts. However, they are capable of being switched to 110 volts (if that's what you currently have).

Now, for me to understand clearly, the junction box that you currently have the Unichlor plugged into, is this a separate 110 volt or 220 volt junction box? And is it tied into the same circuit as the circulation pump?
In other words, when the pump shuts off, does your Unichlorine turn off also?
Or is it constantly powered up?

If it is tied into the pump circuit, then yes, you just need to verify the voltage of this junction box.
If not, then I would recommend rewiring the new power unit to the same terminals that your circulation pump is wired.