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Can You Prove Two Particles Are Identical?

Yes, but isn't that what this hypothetical philosopher ought to be arguing, if he is not already. That it is entirely possible that new things will be discovered that break this well tested theory, therefore you cannot say that you have proven these particles to be identical, merely that as best we know currently, if our theories are correct, then they are identical.

Can You Prove Two Particles Are Identical?

The only thing you've missed is the Eliezer could be wrong in describing electrons as a property of the amplitude flow. Or they could be, but there could be another factor that we have no evidence of until future experiments that 'individualise' the amplitude flows.

If QM is right, which it almost certainly is, and as long as there is no factor connected to the amplitude flow that hides by not interacting in a way we can detect, then Eliezer is right. But if either of those two things turn out not to be true, then he is wrong. And as unlikely as that is, it's possible that that will happen. You can't 'prove' anything beyond a shadow of a doubt. You can only ever say "If we are right about what we are seeing, then the evidence we have fits these conclusions the best. And until such time we find evidence that we are mistaken we shall accept these conclusions."