If the Squid proxy MUST be used AND you cannot fix Squid, then you only have one solution: tunnel the API calls through to a server outside of your network and have that server forward the API calls to Amazon S3 on your behalf.
From a basic view you can just replicate all the S3 calls you use on your external server but you must be aware of the security implications, i.e. restricting the usage of the server to say the external IP address of your Squid server, or even API keys much like Amazon use themselves.
If more flexibility is available try another proxy preferably non-caching like Pound.