Hi welcome to Health care magic forum. Most probably the white particles in the hard stools could be segments of shed tape worm. when the worms are plenty one gets these every time they pass stools, what ever the consistancy may be. some times when the worms are less only hard stools can drag the pieces of tape worm out.
Stools examination will reveal the diagnosis of worm infestation. you are stating that your 6yr old son had gastro enteritis
twice . though the etiology is different, possibility due to worm infestation is also can be a cause. Stomach pain due to worm infestation is a commen symptom.
The tape worm infestation usually occurs in people who take pork in their diet.
I advise you to consult a physician for diagnosis and treat ment, of all the members of the family simultaniously. Wishing you all a quick recovery.