Yes, you can certainly use 'go missing' with objects. If you say 'The diamond ring went missing', it means that it 'disappeared' ( and the implication is that it was probably stolen). Einstein's brain also went missing. And just a bit of language trivia - this expression is one of the very few which American English has adopted from British English in fairly recent years. The vast majority of language influence has been in the opposite direction for the last century, but 'go missing' is a rare exception.

To go missing usually means that someone (or an animal) has disappeared (like kidnapping or death). Gone can also be used but it will be "has gone missing" or "had gone missing". I would think it's more common to say disappear when talking about an object. Either way, I've seen "went missing" applied to objects but it sounds weird, just remember that "missing" is to be taken as an adjective in this situation (the state of the person/object is "missing").