Start by labeling the wires that go to the flippers, you can use Blue Painter's Tape. Then label the coils themselves. You can write on the coil plastic with a Sharpie (it cleans off with alcohol). Now you can put the game back together again without blowing anything up.
Unsolder your now clearly marked wires with a 25w soldering iron. If you don't have one yet, your going to need one constantly to maintain a pinball game. Don't buy one for less than $20, trust me on this.
If there are any "lane change" switches and wires ganged up with the EOS switch, label and unsolder those too. Unlike the EOS Switches, the Lane Change switches usually do not need to be replaced, only cleaned. Just pull a crisp $100 bill through the held together contacts, until the bill pulls through cleanly. The first pull will be very black with carbon.
Loosen the flipper nut, and carefully pull the flipper bat out from the top of the playfield.
Next, get out a 1/4" nut driver and remove the hex head screws from around the flipper bracket. Take the whole assembly out of the game and onto a well lit workbench covered with newspapers. If you don't cover your workbench, you will soon be sorry as the whole bracket will be covered with black carbon and iron dust.
The dust comes from the metal on metal pounding between the Plunger and Coil Stop. Some more dust comes from the spark that occurs at the EOS Switch. It's filthy.
Unsolder the EOS Switch Capacitor, and put it aside.
Take the old Coil Sleeve out and discard it. If the sleeve is tight, press evenly on the bench to get it out. If it is absolutely stuck, the coil may have overheated at some point. Replace the coil, they are only $10.
Discard the old Coil Stop, the Spring, the Plunger/Link assembly, the Bushing, and the EOS Switch. No reason to save them as spares, because once you play on a game with new flipper mechs, set up correctly, you will never even think of reusing that old junk.
Take the saved parts to the sink and scrub with Fantastic cleaner (or any other degreaser) and a toothbrush. Don't get the coil label wet or it will fall off. Just use the damp toothbrush and clean the coil inside and out.
Don't put the metal parts in a tumbler for too long or with high abrasives. The parts are zinc plated and the tumbler could remove this protective plating. When you see restored games where the hardware is completely covered with white corrosion, you know somebody tumbled off all the coatings. If you, or someone before you already did this, tumble again and spray with a light coating of clear Polyurethane.