Probably not, suggests a study funded by Maddie's Fund® and conducted on both sheltered and owned pets in the Pacific Northwest.
In the study, researchers collected fecal samples from 103 shelter dogs, 105 shelter cats, 51 owned dogs and 52 owned cats.
While the shelter pets were much more likely than the owned pets to have Blastocystis in their stool, there was no correlation between that finding and the presence of diarrhea or other gastrointestinal symptoms.
The entire study is available at the link below.
Ruaux, Craig; Stang, Bernadette. "Prevalence of Blastocystis in Shelter-Resident and Client-Owned Companion Animals in the US Pacific Northwest," Sept. 16, 2014. PLOS ONE. DOI: 10.1371/journal.pone.0107496