"Open source" has a couple different related meanings, and it's not clear exactly which aspect you mean, or how it would help here.
- A license which allows the source code to be redistributed under the same terms as it was received? The problem with these boxes isn't that people can't redistribute it.
- A collaborative model for creating software? Possibly -- though there have been some high-profile cases of security issues sneaking into these, too. Modern voting machines also seem to have a huge stack of dependencies, which mean the surface area for exploit is much larger than the voting software itself. Would you require a fully open-source stack, and how far down?
OSS is great for transparency for the person receiving the software (in this case, the jurisdictions operating the machines), but if those people don't care to audit it (or redistribute it to others who wish to), that doesn't buy them anything.
I'd much rather see a rule requiring voting software to be publicly viewable by any voter in the jurisdiction, regardless of the software license it's sold under.