The first thing I'd suspect is the Meike grip...
I've heard and read about enough issues with inexpensive third party grips that I only use Canon OEM and have had very few issues with those (okay, the old 30D and BG-E2 used to loosen up over time, to the point the camera would stop dead in it's tracks.... but all that was needed was tightening the grip).
I've occasionally had the grip's off/on switch get bumped and turned enough that the vertical controls wouldn't work. But that probably can happen with any of the grips.
Even with OEM grip, there may be some minor differences between the shutter release buttons... For example the 7D uses the same shutter button as 1D series on the camera body, with a very short shutter lag. I don't think the shutter button on the BG-E7 grip is the same, but it's an almost imperceptible difference.
I agree with other responses, too... that the Focus Mode must be AI Servo. If trying to use One Shot with moving subjects, you'll miss more than you get in focus. AI Focus isn't really a mode at all... it's just supposed to decide for you whether or not the subject is moving, then use either AI Servo or One Shot, as appropriate. I don't have 6D, but in the past with other Canon cameras have found AI Focus sometimes chooses wrong and tends to add a slight delay that can cause missed shots.
And, yes, Back Button Focusing can be a big help with AI Servo. If not using it, give it a try.
Also, I usually use a single AF point with moving subjects and AI Servo. In particular, with 6D I'd restrict it to the center point only. In 6D all the other points are single axis, only the center one is more sensitive dual axis/cross type point that will be faster and more responsive tracking movement.
OMG, I just noticed B&H is selling brand new BG-E7 for $109... ISTR those originally sold for close to $250, when the 7D was intro'd nearly five years ago. To me that... along with the current prices for 7D themselves... pretty much confirms that we'll be seeing Mark II's of both the 7D and it's grip very soon!