67 352 points at some point starts backfiring through exhaust then quits running

 

First and Foremost; welcome to FTE!! Not an authority, more knowledgeable members will likely chime in with comments:
Suggest conduct a quick timing check to determine if, or, how far it's off. Couple cheap tools I like to use; a button type remote start>http://ecx.images-amazon.com/images/...L._SL1500_.jpg and an inline spark plug tester:http://ecx.images-amazon.com/images/...bL._SX425_.jpg Rotate engine, immediately stop when #1 fires and check position of rotor; if 180 degrees off then go thru 1 more cycle.

A back yard mechanic method I occasionally resort to is remove #1 spark plug, place finder in the opening and rotate the engine until compression forces finger out of the opening, immediately stop then check the position of the rotor, this is only to determine if the timing is in the ball park, fine tune one engine is running. I would also check the idle mixture setting, and float lever, keeping in mind, adjust idle mixture then one needs to recheck timing.

Finally, check for vac leaks, and depending upon Brand, type, model carb will determine whether vac advance operates off primary, or secondary vacuum? Anyhow, suggest wait for more knowledgeable member's to chime-in, food for thought?

 

When's the last time you checked your points for correct gaposis and any corrosion?..... How's your dwell? How's the timing? What do the plugs tell you?

 

Have you changed anything lately? Do you hear vacuum leak at carburetor? How long will it run before it acts up?

 

A weak coil will cause low spark, and not a good burn in the cyl's, so the unburnt gas backfires in the hot exhaust/.

 

Worn upper distributor bushing?