I have a 1979 mercury 40hp that I am rebuilding. The old rectifier was fried and I need a replacement, but I want to know if I have to have a new rectifier hooked up in order for the motor to run.

Also wanted to make sure I run the wires from the rectifier correctly. If I remember right, it was the 2 yellow wires that come from the stator, and then a red wire that goes from the rectifier to the positive terminal on the starter solenoid. Thanks for the help.

Yes it will run but the stator isn't too happy without a load on the AC circuit so don't do it long.
Yes you are correct about the wiring