Research papers have been put out on potential methods to use a public blockchain for voting while preserving privacy and ensuring that only eligible voters vote:

http://www0.cs.ucl.ac.uk/staff/J.Groth/ACNS05VoteProofFull.pdf

https://eprint.iacr.org/2018/466.pdf

Zero-knowledge proofs are at the core of solutions to this problem. They are what makes private transactions on Zcash private.