I sometimes open "Live ATC" app on my iPhone and listen to the "status" of my airplane when I'm taking off until my plane reaches the runway (I know the phone have to be turned off but its exciting to listen to ATC communicating with your pilot). In order to know the status, I have to know the call sign of the plane. If I'm taking off from a small airport, I kinda guess the call sign based on the conversation between the ATC and the pilots and the position of my plane (for example if the ATC says United 220 hold short of Juliet and if my plane is on Juliet, I know my plane is United 220). There have been times when the pilot announces the call sign of the plane before pushback. My question is, Is it okay or weird for me to ask the call sign of the plane to the pilots / flight crew during boarding.
Mostly you wouldn't need to ask. Most commercial flights use the airline's standard call sign and the flight number printed on your ticket. Sometimes the call sign can be confusing - British Airways use 'Speedbird'; PAN AM used to use 'Clipper'. There's a list of common callsigns on http://www.pilotcareercenter.com/Search-Air-Carrier-By-Name-Region-Aircraft-Type-Radio-Callsign#callsign
The flight number should be enough - just listening for calls referring to 'two-two-zero' will almost certainly identify United 220. Code-sharing flights, however, will generally use the flight number of the operating airline. Your ticket might say 'United 220', but if the flight is operated by, say, American Airlines so the call sign might be 'American 406'.
All that said, if you're using your phone when you have been explicitly asked to turn it off the airline is within its rights to confiscate your phone or remove you from the flight.