If you eject I am fairly certain that they will stay, as they did say you could steal enemy vehicles etc but not peoples on your own teams.
That would be my suspicion as well. As Fliyngtiger pointed out as soon as you ejected the neurohelmet connection would be cut, so at the very best any automatic balance systems tied to the gyros would kick in and try to stop the mech and prevent it from falling down. Failing that, it would probably just fall over if the mech were moving with any real speed.
Given the intended ability to be able to steal 'mechs, I believe any abandoned equipment will come to a normal stop, if not immediately, then fairly rapidly and be unlikely to do any real damage. Unless of course, you eject at the very last split second, in which case you're likely to be caught up in the resulting nuclear fireball anyway.