And even if the format only stored the checkbox states as a single bit each (unlikely), compression algorithms don't care. They will behave differently on different byte sequences, which can easily lead to a difference in output length.
Also, it's already been done with voice calls with no attacker-controlled data: https://web.archive.org/web/20080901185111/https://technolog...
The version of your checkbox example which is vulnerable without any formal data compression is when the checkbox is encoded in a field that is only included or changes in length if the value isn't the default, common in uncompressed variable-length encodings like JSON.
Also, zip/deflate etc was not designed to eliminate side channel leakage. Some compression schemes obviously (with padding) can mitigate leaks, but it has to be done deliberately
The special concern with compression is when attacker-controlled data is compressed against secret data because then the attacker can measure the length multiple times and deduce the secret based not just on the length but on how the length changes when the secret is constant and the attacker-controlled data varies. This can be mitigated with random padding (makes the attack take many times more iterations because it now requires statistical sampling) or prevented by compressing the sensitive data and attacker-controlled data separately.