Hi, can piriformis syndrome cause the SI joints to become unstable with its Constance pulling on them. Thanks for taking the time to answer.?

No. No it cannot. It does not pull on the SI joints. These would be separate issues.
No. The sacroilac joints are very stable and the pyriformis muscles not strong enough to disrupt them.
I don't think so. The piriformis syndrome Is a neuromuscular disorders that occurs when the piriformis Muscle irritates the sciatic nerve causing hip and buttock pain. .