TL;DR: No, but if the hashes were collected, one might be able to better tell as to whether or not the SHA256^2 algorithm is broken.
If one can find a way to produce desired outputs from specific inputs, then a hashing algorithm is considered "broken". Both MD5 and SHA1 are know to be broken in this way. Theoretically, running an algorithm over and over again over a set of random inputs (exactly what mining is), could provide insight into patterns produced by the algorithm, thereby allowing one to prove whether or not it's broken. However, this insight would only come through statistical analysis of data gathered, and since most of the data is thrown out -- of the trillions of hashes per second only the ones corresponding to minded blocks are recorded -- this isn't feasible.
Another subtlety: because the algorithm in question is specifically SHA256 applied twice and not simply SHA256, even if the data were collected and analyzed, it may not tell us anything directly about SHA256.