Not a pilot but an avionics mechanic who has worked on several types of aircraft from Jetstream up to 747, and some rotary wing also.
No aircraft I have ever come across, or even heard off, as any kind of auto shut-down system on the engines. If an engine shuts down in flight it is because the crew have decided to shut it down or because there has been a catastrophic failure that prevents the engine from operating.
I am ignoring the possibility of fuel starvation as that is a situation that really shouldn't occur unless human error is involved (BA 38 at LHR excepted).