Can stool softeners give me diarrhea?

Yes. Excess intake of stool softeners can give you diarrhea. Miralax (polyethylene glycol) can cause diarrhea, senna can, lactulose can, even mineral oil in overdose can cause diarrhea. Use the recommended amount.
Yes. Stool softeners increase the amount of water your bowel puts into your stool to try and keep stools soft. If your bowels are very sensitive or you take too much, this can result in diarrhea.