No they wouldn’t. Why on earth would the engines be shut-down?
As Mr. Foyt suggested, there are indeed circuit breakers which can turn the devices off - but it is a federal crime to do so while in flight.
b737.org.uk/circuit-breakers.htm
shows you where they are. [Warning: Not a secure site, but I didn’t catch any cooties].
I believe that the requirement for the CVR (not sure about the FDR) to have an independent backup battery applies to airplanes built starting in 2010
According to www.aviacionline.com - which has a comprehensive run down on those systems - the CVR had a ten minute battery backup (called RIPS) available in the early 2010s, but it enains optional equipment, not required by any laws.
I guess I stand corrected here, so I’d like to amend my comment, changing “cannot” to “shouldn’t”.
I guess everyone reacts differently, but if I’d lost an engine on my airplane, I’d be thinking about something other than how to turn off the voice recorder.
A Federal Crime? Not sure that works outside the USA.