So, what are the possible values for those two equations, given that m is always possitive, g is possitive. The kinetic energy of an object is a measure of the work that would be done if it was brought to a stop, i.e. if I throw a baseball at a watermelon it will crack open.
I think the negative kinetic energy example you gave above was saying that if you throw a baseball at me, and then I hit it with my bat and send it flying in the other direction then the kinetic energy is now negative, but if the ball comes back and hits you it will hurt! This means it had energy! If there is any energy form that has negative energy, it would mean it has a LACK of energy, i.e. it would take work to give it any energy at all.
Kinetic and potential energy can be thought of as bank accounts, raising a ball is synonymous to putting money in the bank (potential energy) that you can use later to buy stuff (do work!)
For kinetic energy, you can think of the speed of the ball as the bank account balance, the higher the speed, the higher the kinetic energy. If you stop it, all the kinetic energy is used up.