Has a SHA256 output ever been found consisting entirely of the same character?
If any filtering criterion on the output of SHA-256 (with its definition independent of SHA-256 internals) leaves $n$ possible values out of $2^{256}$, then as far as we know,


the best method to exhibit an input to SHA-256 matching that criterion involves trying distinct inputs;
the expected number of hashes (compressions) required for this is $2^{256}/n$;
the $n$ outputs are equiprobable, thus the entropy of the process generating SHA-256 hashes passing that criterion from random input is $\log_2(n)$ bit/output, or $n\log_2(n)2^{-256}$ bit/input.


An output consisting purely of one character is such a criterion, with $n=16$ if we read character as hexadecimal digit. It is expected that $2^{252}$ hashes would be necessary to achieve that. Based on blockchain statistics (currently $7\cdot10^{18}$ SHA-256d per second), I guestimate that less than $2^{95}$ SHA-256 hashes have been performed by mankind; thus, with overwhelming odds, nothing giving a hash passing said criteria was ever hashed. Our best estimate of the theoretical entropy rate of this process is $4$ bit/output, or $2^{-250}$ bit/input.

I do not see how the input values found by bitcoin mining could be of any use in the study of SHA-256.