That such a proof must exist in the first-order equational theory of rings follows from Birkhoff's completeness theorem, as I mentioned here recently. Stan Burris told me many years ago that John Lawrence has discovered such an equational proof, but I don't know if it has been published yet. Some clues as to how it may proceed may be gleaned from their earlier joint [work [1]] 1
1 S. Burris and J. Lawrence, Term rewrite rules for finite fields.
International J. Algebra and Computation 1 (1991), 353-369.
http://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/fields3.pdf