"Yeah, but he also said that he thought that because we'd just learned that Julie was pregnant. But that doesn't come till later."
Yes, that's one reason why I've always thought it's a really funny story that Sondheim just made up. Another reason: The opening verse to the song, in which Nettie sings about March, April, and May, very clearly sets up that she and the chorus are about to sing about June. So even if the diction of the singers wasn't great, I would think it would have been very clear for that reason that they were singing "June is" rather than "Julie's."