GPS aviation units, generally, can not interfere with A/C electrical systems. Something about the circuitry design. However, ALL other electronic devices do generate some kind of electrical signal, even if that device is only a receiver. If you happen to be sitting in the wrong spot, this signal MAY set up some kind of harmonic reaction in nearby wires. (Lots of wires in an aircraft!)This could cause the wire to become, in effect, an antenna. Not very powerful, and certainly not enough to be classed as harmful radiation.
However, this wire is now an antenna for the electronic device, as well as serving whatever use it was designed for. Results are unpredictable. Slim chance of interfering with A/C systems, but there is a possibility, however slim.
Remember something else. If someone wanted to build a resturant or bar using the same design as an aircraft, with huge quantities of fuel and electrical systems in such close proximity to the customers, it would not be allowed. Too much risk!
Now, do you really want to take chances, however slim, at 35000ft with something deemed a hazard when on the ground?
The Captain is the Boss, and what he says, goes. Even if he is ignorant, ill mannered, and wrong, he is still the Boss. If you can't accept that, take a train or ship instead. Turning anything on 'in secret' shows that you know you shouldn't, thats WHY you are being secretive. Might make you feel smug, but go play your games somewhere else.