[psi-squared]

I am not a security researcher, but I think you could keep the benefits of both compression and security, as long as you're careful on the server side:

Say you have a document structured like [boring data] [secret data] [boring data]. I don't know if any existing compressor lets you do this, but the gzip file format (really the 'deflate' format used inside it) allows you to encode this (schematically) as follows:

[compressed boring data] || [uncompressed secret data] || [compressed boring data]

where each || is i) a chunk boundary (the Huffman compression stage is done per-chunk, so this avoids leaks at that level), and ii) a point where the encoder forgets its history - ie, you simply ban the encoder from referencing across the || symbols.

If you wanted, you could even allow references between different "boring" chunks (since the decoder state never needs resetting), just as long as you make sure not to reference any of the secret data chunks.

Edit to add: Also, if the "boring" parts are static, you can pre-compress just those chunks and splice them together, potentially saving you from having to fully recompress an "almost static" document just because it has some dynamic content.