Unfortunately no you can’t do this with the standard box. Both the wave engine and control 4 would be driving the button box with their 5V supplies and could damage one or the other. It might be possible if you disconnect one of the 5V pins on one of the 2 GX12-6 cables. That way only 1 device is sourcing the 5V to the resistor circuit. But there might be issues due to parallel impedance of 2 separate inputs, since the resistor tree is designed specifically to the hydros 0-10V input structure. If you leave a hydros 0-10v disconnected you’ll notice it reads 2.48V, this is due to an input bias/protection circuit on the input. And the resistor values used in this design were specifically picked based on that input circuit.
so after talking this through I don’t think it would work.
What you could do is connect this button box to your hydros control 4 0-10v input port. Then buy 2 hydros APX cables. Connect one to your hydros control 0-10V output and the other to your wave engine 0-10V input. Then get a RJ45 coupler from Amazon and connect the 2 APX adapters together. Then setup an output on your Hydros Control 4 to output 10V when a mode is activated by the button box. Then on the wave engine set up your feed mode to trigger on a 10V input. It’s a round about way but it’s one way to link a control 4 to a wave engine. You could also use a direct GX12-6 female to female cable (like I sell with my box) but again you have to disconnect the 5V wire between them to prevent the driving issue above).