Everyone interested in productivity has read GTD... it's got a strong following.
Personally I only use some of the concepts, because I find it isn't suited to the way I work and lifestyle I lead. I think it works better for people who have a more structured life. It is an extremely structured way to organize yourself.
The biggest takeaway i got from it is this:
"things that you don't deal with take up space in your mind and slow you done."
This can be anything in your life. Not necessarily business. So, if you had an argument with your wife and left it unresolved, it takes up mental space subconsciously and will reduce your productivity. even if you don't realize it. If you have 3 tasks you are procrastinating about, they will take up space also.
The way the author deals with it is to schedule them in a system so that he can 'forget' about them - it only works if you have 100% trust in your system, so that you can forget about them and know they will get taken care of.
Read the book. You'll definitely get value from it.
First point: The static stretching usually used by people is scientifically proven to only do damage and weaken your muscles and performance. Please don't encourage your employees to do that.
Having said that your idea is good if you can get people involved. It would provide community building / teamwork benefits as well as health benefits.
Incidentally, there is a lot of research also to prove that exercise and greater fitness improves your energy level - so it is a great enhancer of productivity. An excellent book to read on the subject is "The Power of Full Engagement" by Jim Loehr and Tony Schwartz. I highly recommend it, 5 years ago it transformed my approach to productivity- not just in work, but in life. I've never looked back...