For fixed $n \in \mathbb{N}$, Birkhoff's completeness theorem implies that such a proof must exist in the first-order equational theory of rings - as I mentioned here in a recent [post][0]. Many years ago Stan Burris told me that John Lawrence discovered such an equational proof that works uniformly for all $n$ (possibly also for Jacobson's form $x^{n(x)} = x$). I don't know if the proof is published yet, but some clues as to how it may proceed might be gleaned from their earlier joint [work \[1\]] [1]

[0]: https://mathoverflow.net/questions/30220/abstract-thought-vs-calculation/30273#30273

[1]: https://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/fields3.pdf
 
[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