The infamous Canon (A-Series only) shutter "squeal' is not caused by a malfunction in the shutter
mechanism, but by a problem in the mirror cage. The assembly that causes this is the Mirror
Retard Gears /Inertia Gear Train ( Canon calls it the-Mirror governor) . Cameras have been so bad the mirror rises slowly taking 2 to 3 seconds (or more) before the shutter releases and the mirror returns to viewing position also very slowly.

You can refer to Canon AE-1 Repair Guide: for drawing. However, the text only states . "Check mirror governor for smoothness and quietness" Since no mention is made of repair in the Canon AE-1 Repair Guide, you should assume that replacement was the factory cure. (When a camera is disassembled for a complete overhaul the camera unit is oiled by most camera technicians so as to eliminate the 'squeal'). Refer Fig. No. 2-26 page 25 in this section for short-cut 'fix' to this problem (requires minimal disassembly). Also the Canon AE-1 Repair Guide (Factory Manual) and/or Photos Figs. 13, 15, 66 & 74 on the 3 pages From PREFACE of the Canon AE-1 Factory Repair Guide 

Fig. No. 2-26

Figure 13

Remove the Front Decorator cover retained by 4 screws, 1 pointed out in Fig. 13 & 15  at beginning AE-1 Repair Section. What you are attempting to do is oil the gear pivot holes in the Mirror Governor/ Mirror Dampening Gear train.

Figure 15

What is needed is a Plastic Tube Hypodermic syringe using a Needle with a 2" length stainless steel
removable needle, small diameter with a pointed tip, not blunt, but one as used in hospitals and
doctors offices to stick you in various places while stating this will only feel like a gentle pinch. You
will need to alter the needle by going back from the tip about 3/4" and bend this 3/4" section down
to about a 45 Degree angle. Do not be concerned if needle flattens slightly at bend as you are only
going to need a very small opening. Place the altered needle on the tube and test to see if needle will
expel air or if bending closed completely. If closed completely you can try to save by crimping
inwards the sides of the bent portion slightly. It would be best to have several extra needles in case
you need to reform another or over bending breaks your first needle. Once you have a good needle
use a permanent marker or like, mark top of needle in several places back towards the tube or on
tube, with bent part of needle pointing straight down. These marks are for use as alignment when
tip can not be seen, described below.

Obtain a bottle of generic Clock/Watch Oil, NyOil or a like type of fine mechanism oil. Refer to first paragraph and also remove the top left Lens Bayonet screw, What this does is give you external
blind access to the Mirror Governor.

At this point it would be to your advantage to have or have access to a Front Panel/ Mirror box assembly from a junk camera for testing though with good judgement of small distances and a good 'feel', you should be able to complete the lubing with no problems.

 Photo 74 shows gear train with Mirror Control Unit (Canon-Mirror Mechanism) removed. Note the flywheel on top which has a small gear behind it. This is gear you are trying to 'feel' with pointed tip of your needle. With no oil in your syringe, insert the needle into a screw hole, do not force but alter the length and/or degree of bent portion until you can insert it easily. With mark/marks you made on top of needle facing up so you can judge where pointed tip of needle is, try to locate the gear mentioned above. When you are sure you 'feel' the gear, spin it
slowly several times with tip of needle inserted in several of the 'V's between gear teeth then
withdraw the needle.