I've been working with binary BCH codes (63,39) and (63,45) for several weeks, where an additional even parity bit is added to the data to create a 64-bit codeword, which is modulated and transmitted to a distant receiver. The distant receiver is able to us that encoding to detect and correct transmission errors.

Question: Will BCH codes work with non-binary symbol values? For instance, instead of n=63 for GF(2), could a (31,21) code be used with 2-bit symbols? If so, are there advantages or disadvantages to using 2-bit symbols instead of 1-bit symbols with BCH encoding?

Yes.   Reed-Solomon codes are BCH codes typically with 8-bit symbols instead of binary.

That's the fun part about abstract algebra; coding systems based on finite fields use the same operations regardless of the underlying symbol representation.