Someone told me over Yom Tov that one year, they had an interesting thing happen in his shul. On Rosh Hashanah, a man walked in during davening. Not recognizing him, one of the gabboim went over to welcome him, and ask if he needed anything (tallis, machzor, etc.)
The man replied that he was not Jewish. He believed that Jews are the chosen people, and followed the sheva mitzvos b’nei Noach. He believed that Hashem judged everyone on Rosh Hashanah, and therefore wanted to join in davening so he should have a good year.
I believe they allowed the man to stay for davening. I just thought this was an interesting thing.
My brother told me that a day before Yom Tov, a non-Jewish coworker asked my brother if he (the coworker) can also throw bread in the river so he can get rid of his sins.
Maybe he was just plain crazy?