Sorry about the one-liner, I was working at the time. The answer is probably in config.sys. The test image and Volumio will not have the same config file. The rainbow square is very early in the boot process, but the config file is used by the code embedded in the Pi.
See typical boot sequence described here:
http://www.runeaudio.com/documentation/troubleshooting/rpi-troubleshooting/
What I was trying to get at was, that if I were going to diagnose this, I would disconnect everything except what I needed to use to log in and edit config.txt.
Everything connected to the USB port, everything except the power supply and what I needed to log in. I would start with a fresh new SD card if you have one, flash it with the SAME image, but do not change the config.txt yet, leave the DAC connected to the 40-pin header and power, and boot with the original config.sys, with dac driver disabled but dac connected. This should work, if the test image worked.
Then change one thing at a time.
I would disable the turbo mode in the config. sys, and the USB high current limit and boot again (neither is necessary to get a dac running, and I bet the test image does not use turbo mode or the higher USB current setting).
After that, make any HDMI settings, and boot again (still leaving the DAC dtoverlay section alone.
Then set up the network, etc...
Then any other Rune Audio settings, but leave the output set to HDMI.
Only at the end, uncomment the line in config.sys for the dac, and try to boot.