Relay bypass is very dependent on what are you are connected to – if there is any DC from either the thing before it or the thing after you will get a click as it is interrupted.
Not much you can do without changing your rig – a relay bypass is just like a switch, and switches can click. My guess is that it is your active pickup – see if you get the same click with the guitar disconnected/switched off. Assuming your DI is a transformer type, it is unlikely to have any DC.
It is possible that for some reason there is DC on the output of the Eclipse – try disconnecting your guitar, then see if you get the same click when you plug/unplug the Eclipse output.
