You can't simply adapt an ethernet cable to output music. You need some sort of device that decodes the data sent over the ethernet cable to music. Your Squeezebox seems to have served that purpose.
A small computer like a Raspberry PI could do the trick. But they don't have SP/DIF outs, so you will have to convert the HDMI output to whatever output format you prefer.
But to answer your other question: Yes, the Cat6 cable (and even Cat5) should "pass" 24 bit files just fine.