How to test a CPU watchdog on board?
You can test through a sequence. Use an existing non critical output like an led as a test signal. Program the board with a test sequence that will toggle the led and loop, 
And does not pet the watchdog. Test for the loop. Then program it to do a second loop,  where it will not reach if the specified watchdog timer does not reboot. Test to make sure it passes the watchdog petting. These two steps will make sure the hardware part of the watchdog are good. 

The rest is ensuring your final code does not affect the watchdog by disabling it or petting it when something that shouldn't lock up gets locked up. 

As to your question about the external clock being stopped, I'm not familiar with any microcontroller that would have two independent clock sources that would keep the watchdog ticking when the main clock source is disabled. In a externally grounded clock situation, a watchdog reset wouldn't fix that anyway, unless your code can switch to an internal clock for the main code after that reset. The watchdog is meant to clear software related hangs.  But I'm sure those exist for mission critical applications.