Do you ever use yoga postures to stretch out your personal training clients?

Yes I do, but I am a certified Yoga instructor as well as a trainer. I have a few short pose flows that I use at the end of a session. Supine stretches work very well together and flow nicely. for example with client on their back have them draw knees to chest ( a bundle pose), then w/ feet on the floor they can lower knees to one side then the other holding on each side for at least 5 - 10 breaths. Then Bridge pose, then supta budaha konasana, (cobblers pose, but lying down), then they can roll over to the right side take a few breath then come up to seated. I usually take that time to talk about our next session.
Cheryl P Murman

Yes, I include Yoga movements with my personal trainer clients

Sure! Funny though, I don't always remember the origin of certain movements i.e., is the cat/camel something I learned in yoga class years ago (yes), or in a Stu McGill workshop (yes)....just from different perspectives. It's a fantastic move for spinal mobility. Other postures I regularly incorporate into my clients' sessions include:

-downward dog
-up dog
-modified cobra
-child's pose
-triangle pose

I do. I even use them with classes.

I find that some of the yoga poses take on different names in pilates.

I tend to put more emphasis on the function of the movement and what muscle groups one is stretching over the name of the movement.

I think it is important for our participants to understand the why behind the chosen stretch.

I use them in my classes sometimes. I am not a certified yoga instructor but I feel that we can all benefit from the way some of the "poses" stretch and move you. And, I agree with Karin and Joanne that it is more about the function of the movement and I use what is familiar so that I can mention what it is benefitting and how to properly execute.