I use a Saffire Pro 40 (driver version 4.1.4.18046, dated 1/16/2014) and Mix Control 3.41b. I have not seen this behavior.
I doubt it has anything to do with the Mix Control application, which does not come into play until you explicitly invoke it. I suspect it could even be uninstalled and the audio interface would still function, just without the convenience of a software UI. It certainly has nothing to do with starting or stopping the audio engine.
The engine will shut off automatically if for some reason it can't fill the audio buffers for 250 milliseconds. For example, a background process that won't let go of the audio subsystem or an interrupt handler that won't allow Firewire interrupts to be serviced. That might be an angle to explore.