I'm not exactly sure whether they intended those two connectors to be used with a traditional 3-5 volt (or higher) Solid State relay directly or not... I think I recall those two being an actual internal relay... but only good for a very small amount of current. Did you download the hardware manual and see if it tells you anything ?
I've simply just used the output terminals. I am not sure of your setup, but if you have any open pin on the outputs plug, you can easily use that to trip a solid state relay.
For example, if you have, say, output pin 4 available, connect it to the positive side of a 3-5 volt input solid state relay(SSR). Connect Flashcuts Output GROUND or COMMON from the plug to the negative side.
Now, you will need to use a SSR where the OTHER terminals match your needs to turn on and off the cooling pump. If the cooling pump is just a 110 volt pump, you use a SSR that can switch something in that range like this one:
https://www.amazon.com/dp/B00PQ7Y8ZU...a-370268697797
In the case of 110v, you simply interrupt the Line connection with the SSR so the relay can turn on and off the power just like any switch would...... You find the wire designated as the Feed or Hot Line and connect it to one side of the high voltage side of the relay, and then provide that Feed or HOT connection from your source power to the other connector.
Then you go into the OUTPUTS setup of Flashcut and define an M-Code you can use to turn OUTPUT LINE 4 on and another M-Code to turn it back off. Their will be dropdowns to indicate which output you want it to use.
FINALLY, in the G-Code, you put the M-code you defined early in the code so it turns on, and the M-Code for stopping it near the end of the code. Of course you will want to configure your CAM programs post to automatically put it where you want it.