Hardy & Wright have an elementary g.f. proof (but not very short). There is a slick two page proof by Mihael Hirschhorn, with a revealing title "A Simple Proof of Jacobi's Four-Square Theorem" (available here). It uses only Jacobi triple product identity which itself has several short proofs (see here and bijective proofs here). A combined proof is accessible to undergraduates.

P.S. There is also a curious proof by Andrews and Zeilberger: A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares. The story behind this proof is also interesting, but I can't remember if it's written anywhere or just something George Andrews told me.

Hardy & Wright have an elementary g.f. proof (but not very short). There is a slick two page proof by Michael Hirschhorn, with a revealing title "A Simple Proof of Jacobi's Four-Square Theorem" (available here). It uses only Jacobi triple product identity which itself has several short proofs (see here and bijective proofs here). A combined proof is accessible to undergraduates.

P.S. There is also a curious proof by Andrews and Zeilberger: "A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares". The story behind this proof is also interesting, but I can't remember if it's written anywhere or just something George Andrews told me.