"Will my shares continue to show up in my brokerage account until something happens with Barz? ... Even though Barz is now a private company?"
Probably not for long. They're outside the purview of BDs at this point, so they should disappear from your statement. If you're lucky, they'll send you an explanation.
***** So how do I have proof of ownership? What happens when/if Barz goes public again, will shares automatically show up in my brokerage acct?