maylit,
There is no such thing as a "beejeeber". The dictionary lists an older form "bejabbers" It is a mild form of swearing. "Bejeebers" or "bejabbers" is a euphemism for "beJesus".
He scared the bejeebers out of me.
She knocked the bejeebers out of him with one punch.
It is similar to saying "hell" as in "He shook the hell out of it." or "It scared the hell out of him."