This is a temporary solution, at best... but it will work for a while.
When the A-series Canons develop the dreaded "shutter squeak", it's actually sort of a cry to "please clean and lube me properly throughout!" It actually isn't related to the shutter, but is a lack of lube in the mirror mechanism that causes the noise.
However, the trick with the syringe works. Let's see if I can do this from memory... Oh wait, I just Googled it and found this illustrated site that explains it at least as well as I could!
Notice the "mirror governor" in the drawing near the top? That's what you are trying to get a tiny drop or two of oil onto. I use a lightweight oil that's used by clock makers, similar to "sewing machine" oil.
Working through the screw hole with a syringe to lube it, you are doing it blind. You can't see the mechanism you are trying to get the oil onto. It's also possible to oil this from below, after removing the bottom plate of the camera... or from above, after removing the top. From the top, I think you can see some of the mechanism, but you have to do a lot of disassembly. Through the hole is the quickest and easiest. I've never tried it from the bottom, learned to do it through the screw hole and probably have done this quick fix on a couple dozen A series camera over the years.
If you get too much lube in there, it can run into other areas you don't want it... onto the mirror or shutter or up into the pentaprism. Only one or two little droplets are needed. I suggest practicing with the syringe outside the camera first, to see how much oil is pushed out with any particular amount of pressure on the plunger.
But if the camera is squealing like that, it's really due for a proper, complete CLA or "Clean Lube Adjust". If you were quoted $120 for the fix, that's actually a full CLA, in which they would disassemble, clean thoroughly inside and out, lubricate mechanisms throughout the camera, check shutter accuracy and adjust if needed, check the metering system and adjust if necessary, replace light seals in the mirror box and around the film door, possibly other places (varies in different camera models). You might be able to get this done for a little less than $120 if you shop around, but that's actually about what I'd expect locally. After a CLA, the camera should be ready to shoot for another 25 or 30 years or so. So that works out to, what, $4 or $5 a year maintenance? Really not bad, when you think of it that way.
A freshly CLAed camera will usually bring a higher price from a knowledgeable buyer... but probably not enough to recoup both the purchase price and cost of the CLA. About the top price I've seen for an A-1 in good condition is $100 or a little more. But I really haven't been tracking prices of them lately.