The compounding of "Be quiet" and "Shush", normally used when you really want a person to SHUT THE FUCK UP!!!!!!. For more oomph, the word "a" is put in front of it.
"BeShush damn you, your so annoyin'!"
"I said a BeShush!!!!!"
Beshushka is the code name for the famous Czech alcoholic drink Becherovka. For some people Becherovka is too hard to remember so they just call it Beshushka. This drink has brought good times to many people and it is deeply revered.