The Point
Research clearly shows roughly a 20% increase in work related "activity" as a result of BYOD. The real question is whether or not this is "productivity." The key factor is whether or not it generates good, crisp business-grade work or if it just makes people feel more connected and/or responsible for their work, which isn't all bad.
As for those who can't see the "benefit" the day will come when "flexible schedule" will mean 24/7/365 and that will be the expectation. In many cases it already is the norm... And, frankly, for many IT pros it's been the norm for a very LONG time... The rest of the business is just catching up.
I've been the Disaster Recovery on-call 24/7/365 for a company for years, with a DR phone within reach at all times. It takes it's toll. You are never really "off" of work. The price is paid in not sharpening your saw. That is why activity and productivity need to be segregated for the BYOD research to be relevant.
In many cases businesses are shooting themselves in the foot by encouraging tired and dulled workers who really could use some vacation to sharpen up and bring their A-game to work every day.