Sounds like what you're looking for is an ATX12V plug. It's a 2x2 connector that normally has two yellow and two black wires. It normally goes into the 2x2 receptacle on the motherboard to power the processor. In this case, the eGPU enclosure needs it for some reason, maybe for more power.
The good news is that the 2x4 breakaway connector (called EPS12V I believe) that splits into two 2x2 connectors is probably compatible with this receptacle. One of the two 2x2 pieces of the connector should fit into the eGPU's power receptacle, and the other won't. If it fits, it is probably the right connector. If two of the wires going to that connector are yellow, and two are black, then it's almost certainly the right connector.
You may have multiple of these 2x4 breakaway connectors. If so, they should behave identically, and you can break up any of them and try to fit the pieces into the ATX12V receptacle.
List of ATX power supply connectors: https://en.wikipedia.org/wiki/Power_supply_unit_(computer)#Connectors (without images, unfortunately.)
Don't forget about the big 20-pin or 24-pin main ATX motherboard power connector. Your second power supply, since it is non-modular, will need something to simulate the motherboard's power button. That's can be as simple as a switch between the PS-ON wire (green) and any ground wire (black). But hopefully your eGPU has a place to plug in the ATX motherboard power connector, and handles that on-off switching for you.