Can budgies get the poop scared out of them?

That seems very possible. When budgies get stressed/scared, their poops can be very watery. It happens to our Penry on occasion. The fact that Bertie's droppings normalized makes me think there was a stressful period when the storm was happening that made his poop watery. If you think about it, this happens to humans too-- when humans are anxious/nervous/stressed it can be reflected in their poop!

Sometimes, I even think Penry will get a little over-excited and drop a poop that way. If he sees a bath, or gets let out of his cage after a while inside, he will do a funny little dance and poop immediately. Crazy birds!

Thanks. That's kinda what I suspected but I'm never sure when it comes to bird anatomy. Also there sneaky hiding It when they are sick

Yes stress and excitement are sure to produce Poop.
the first thing Budget does when let out is fly straight to my shoulder and you guessed it !