Can you really write with no hidden meanings at all, or will everything have an undertone even if unintended?

This is a tricky one, because there are actually two things at play here. Writing and reading are two entirely separate actions, so the writer’s intentions are only part of what the reader experiences.

As a writer, I have written what I thought were straightforward stories, only to have people in my critique group pull out details and assign them meanings I didn’t intend. There’s an entire school of hermaneutics (literary interpretive theory) that says that interpreting the written word is as creative an act as the writing itself, and that ultimate meaning of any text rests with the reader, not the writer.

So, it really isn’t possible to write with no hidden meaning. Even if you didn’t hide anything in your text, your readers will supply it.