>Can a non-believer have a mystical or spiritual experience?<
I don't see why not. I think we all have similar experiences, it's just our explanations of them that are different, depending on our frames of reference.
If, for example, at one place, a female "otherworldly presence" was experienced by a lot of people, a Pagan may well say they'd encountered the Goddess, a Christian may say they'd encountered the Virgin Mary, a Buddhist could well consider they'd connected with Kwan Yin and an atheist could have concluded that they'd felt something that seemed very feminine.
All of them had a very similar experience, but for each one, the interpretation differs according to their own self-imposed beliefs/comfort zone.
This is why describing and talking about subjective experiences is so difficult, we don't have any common reference points or adequate language to communicate what's happened outside our own frames of reference and we tend to either generalise or use flowery or vague language. Maybe these things don't translate into words.
As for my own experiences, I don't know where they're generated, whether they are my brain reacting to outside stimuli at sites although they do happen in other places too, or whether it's something else, a force or "the universe" or whatever. One thing's for sure, whatever it is, whatever label anyone wants to put on it, it's more than incredibly pleasant and often profound, I'm up for it anytime.