B&S 344cc engine sputters/dies after 20+ minutes

One of two things:
 
Either the ignition coil is failing, which is not un-usual..
 
When it dies, have a known good spark plug handy.  Plug it into the 
spark plug wire, ground to head and look for spark..  
 
Also, do you have a low oil shut off switch on that engine?  It'd have a thin, probably black,
single wire going to it.  If so, those are known to fail (especially on generators). They kill most 
of the spark...    
IF you do have it, just un-plug it to by-pass it.  It won't hurt anything but won't be a "safety" should you let your oil level go low...
 
Both of those cases either kill most or all the spark.   
 
Posting a full engine model # of the engine would help check the parts list to see if 
your engine is equipped with the switch...
 
.Please post back what you find..
 
.