Short answer: Yes, proof should be required.
I struggle with the argument: If we as a society require proof of citizenship, then we as a society should provide the means of proof of citizenship. I.E. at birth, provide with a free birth certificate from the county courthouse (or free form of whatever identification would suffice) - whether a home-delivery or not, etc.
The reasoning behind that argument is that such a new law would be imposing additional hurdles to vote (which is an inherent right of all citizens) and that we should never obstruct inherent rights, unless they infringe on the rights of others.
That being said, there are obstacles to vote. You must have the means to get to the polls for example. We don't expect our society to jump those hurdles for each individual, but then again, those hurdles weren't created by society.
Regardless, my answer is 'yes, you need proof'. But how to get there is a question unanswered.