An EZ-Out is a type of broken screw remover, of which there are several. For small screws, I prefer a left-hand twist drill bit, as you drill it has a tendency to unscrew the bolt. It is very important to center-punch the bolt where you are going to drill tho, cuz otherwise the bit will wander around.
If all of this is beyond you, then take the part off and take it to any machine shop. It's a five minute job to anyone who has the right tools and know-how. Minimum shop charge or even free if they are in a good mood.