Is it safe to run/jog with a Flash drive Macbook Air? I want to be able to put an 11" Macbook Air into a backpack and run to work/gym with it. Is that safe to do that, given these devices have no moving hard drive parts? What about the other components, would they handle shaking?

Bonus points: is there any backpack made specifically for doing this?
There are definitely no moving parts. I wouldn't worry about jogging with it. I wouldn't even worry about jogging with a non-flash drive equipped laptop if it was turned off. The drive spindles and heads are immobile when the drive is powered down.

For a pack, look for something that suspends the laptop away from the sides. Preferably holding it in something that'll absorb the shock a bit.