From what you described sounds like your rear struts are shot. Most likely the rubber boots around the struts are so badly cracked that the compressor can not even begin to fill them.
That is why it runs for 5 minutes as you described and shuts off.
If you can have your rear of the car jacked up and inspect the struts for any wetness around the bottom of the struts. This would indicate that they are leaking oil and need replacing.
As for the rear 'haunted house' squeak, it could be a few things going on. Could be bad upper strut mounts....bad center sway bar bushings.....springs could be needing a lube on the bottom where they are mounted to....could be an old wooden door with old hinges in your trunk moving when you go over bumps....
Anyways keep us informed on what you find so we can help as needed.
Are you having the front strut bushings replaced as well? good idea to.
Oh and just for your information.. Bonneville'* technically have air shocks in the rear. Just so you don't confuse the parts guy when you order them...lol
If you order new rear shocks you will need the air line kit part #AK29 found at most parts store for around $8
Good place to order our parts is found at:
here is the discount code link found here: