That's known as the "diff bracket" to most of us but it may have a more official name. I have a spare bracket you can buy off me if you need one. PM me if you're interested, I only have one side so I would need to confirm if its the drivers side first.
let me ask you the mechanic said when he took the screw out the rubber piece came out with it. is the rubber a separate part or is it permanently attached to that bracket? how much do you want for that part
thanks adam