I'll do my best without quoting the complete maintenance manual or Min Equip List.
1.) There is only 1 (one) autostart switch, which enables the automatic starting of all 4 engines.
2.) Each engine has one "Start switch"(overhead panel) and one "Fuel control/cut-off lever"(pedestal-behind thrust levers)
3.) To accomplish an 'autostart' the Autostart switch is selected on and for the engine you want to start you pull the start switch and lift the fuel cut-off lever to 'run'. This gives the engine EEC/FAFC control authority over the starter shut-off valve and the fuel control unit to open/close the start shut-off valve and to command "fuel-on" when the EEC/FAFC sees fit.
4.) A manual start (autostart switch selected 'off') is pilot controlled- i.e when the pilot pulls the start switch the starter is engaged and when he deems an appropriate time (based on EGT/RPM) he selects the fuel cut-off lever to 'run' then fuel/ignition is introduced to the engine being started. The pilot must be vigilant to monitor EGT to avoid a 'Hot/Hung start' and it is up to the pilot to move the 'Fuel Cut-Off Lever' to cut-off if one of these conditions becomes apparent.
5.) The above scenarios ( 3 & 4 ) relate to ground starting of engines and with the powerful P & W APU fitted to the 747-400 even two engines can be started at one time. To start engines 2 & 3 for example you do the following- 'Autostart switch ON' '2 & 3 Start switches PULLED' and '2 & 3 Fuel Cut-Off switches to RUN'. Engines 2 & 3 EEC's/FAFC's will then take control of start valves/fuel controls and 'Bob's your uncle' the engines will miraculously start without further pilot input. Hot/Hung starts in this case are monitored for by the EEC/FAFC and the big computer will cut-off the fuel if it deems one of these events is immenent.
6.) With the aircraft on the ground the EEC/FAFC will carry out 2(two) start attempts on an engine and if unsuccessful in achieving a start will discontinue and spit up an EICAS message.
7.) If the aircraft is in flight and an engine has flamed-out or been shut down, the following two scenarios will play out
A.) If Airspeed/Altitude is within the 'Inflight start Envelope" - when the 'Fuel Cut-off Lever' is placed to "RUN" the EEC/FAFC will introduce fuel/ignition as it determines to accomplish a successful start. It will cycle fuel/ignition ON/OFF indefinitely until the engine starts (unlike on the ground where only 2 attempts are made)
B.) If Airspeed/Altitude requires starter assistance then a prompt will appear on upper EICAS stating "ENG X-BLEED" which will prompt the pilot to pull the start switch for the engine in question and thence the EEC/FAFC will commence an indefinite number of start attempts but will also control the engagement of the engine pneumatic starter in addition to controlling the fuel ON/OFF command.
8.) 'Flameout Protection' is a different kettle of fish altogether and nothing really to do with 'Autostart'. Once an engine is running (For all intents and purposes this is deemed as greater than 50% N2(P&W and GE) or 50% N3 for a Rolls Royce) the automatic flameout protection is enabled via each engines EEC/FAFC. This is accomplished via a comparison of N2 vs compressor discharge pressure. If the ratio suggests a flameout or reduction of RPM below minimum idle (approx 60-62%N2/N3) then the EEc/FAFC will command both ignition systems "ON' regardless of whether "SINGLE" or "DUAL" is selected on the overhead panel.This 'Flameout protection' is active at all times an engine is running and the autostart position has absolutely no bearing on this function. In fact the early -400's delivered to my fleet didn't even have the "Autostart" fitted as it was an option back then. We have since retrofitted our earlier aircraft to achieve commonality with our later deliveries where 'Autostart' was a standard Boeing fitment.
9.) To the original question - if for example one particular engine developes a problem with its autostart feature (an EICAS to EEC interface problem for example) then as there is only 1(one) Autostart switch the whole autostart system is inopped. This is to reduce the chance of confusion for the crew if an engine was to flameout during flight. They wouldn't have to refer to the Tech Log/ MEL to remember which engine out of the four it was that won't autostart. Just inop the whole lot and then the crew know that they are obliged to carry out the engine start sequence manually.(as per para 4)
Hope this answered your question but as per usual it is hard to get 300 pages of Boeing training material/Maint Manual into some sort of concise post. Feel free to ask more(even specific) questions if this is not enough.
Spanner.