Actually, the 340-500 (and I presume the 600, and in fact the 300 as well) instruct you to LAND ASAP in the "After Eng Shutdown" checklist. FCOM ref 3.02.70 page 4.
There is a further proviso warning operators not to windmill the engine beyond 10 hours if the "ENG OIL LO
PR" comes on.
After losing one engine on a four engine aeroplane the skipper has a fairly liberal interpretation of what constitutes a "Suitable Airport" for the Land ASAP instruction. Destination, once en route is perfectly acceptable if that is operationally feasible.
After losing two engines, of course, Land ASAP takes on an entirely different connotation. "Suitable Airport" is really the closest suitable airport. And no, the performance isn't that brilliant on two engines at anything approaching all up weight, particularly if you've lost them on the same side.
For what it's worth, Boeing's definition of "Suitable Airport" for their LROPS can also be Destination on a twin engined aeroplane, if for example, the other options involve landing at Siberian en-route alternates.
I think you have to make your own decision at the time. Was it just a precautionary shutdown due to oil pressure and such, or has it been the result of suspected severe damage? I don't know that I would have continued after I suspected the engine had suffered damage. The only hard and fast rule that springs to mind is to find somewhere to land ASAP after an engine shutdown due to fire.
But I wouldn't get too worked up about the whole thing. When its your turn, you will inevitably upset a lot of armchair experts and seat polishers anyway. Whatever you decide to do, just make sure you get to walk away from it.