I had a very similar UFO sighting, just a couple months ago. Fortunately I've been consuming rationalist media for a long time, and I was able to say "There is a non-magic answer to this question, just because I don't know the answer doesn't mean UFOs exist. My map is incomplete, but the territory isn't magic."

It doesn't make the creepy shiver-up-your-spine and cold-knot-in-your-stomach feelings go away, those are biological reactions. But it does let you accept them and ride them out, like the cramp you know will go away in a while that isn't ACTUALLY a knife in your leg, no matter how much it feels like it.