Not OP, but 'zipping and encrypting' one thing (a file for example) does not leak information by itself. The problem comes when an adversary is able to see the length of your encrypted data, and then can see how that length changes over time - especially if the attacker can control part of the input fed to the compressor.
So if you compressed the string "Bob likes yams" and I could convince you to append a string to it and compress again, then I could see how much the compressed length changed.
If the string I gave you was something already in your data then the string would compress more than it would if the string I gave you was not already in your data - "Bob likes yams and potatoes" will be larger than "Bob likes yams likes Bob".
If the only thing I can see about your data is the length and how it changes under compression - and I can get you to compress that along with data that I hand to you - then eventually I can learn the secret parts of your data.
So if you compressed the string "Bob likes yams" and I could convince you to append a string to it and compress again, then I could see how much the compressed length changed.
If the string I gave you was something already in your data then the string would compress more than it would if the string I gave you was not already in your data - "Bob likes yams and potatoes" will be larger than "Bob likes yams likes Bob".
If the only thing I can see about your data is the length and how it changes under compression - and I can get you to compress that along with data that I hand to you - then eventually I can learn the secret parts of your data.