that's a good question, and I think it helps to know that energy is a relative quantity. Basically, we choose what 0 energy is, and then express the energy of a system with respect to that.
In the example you give here, the 0 point energy is taken to be the energy of the system when the electron and the proton are not bound together, sitting in place very far away from one another.
The fact that the energy of the hydrogen (i.e., when the electron and the proton are bounded together) is negative, means that this energy is lower than it would be if they were separated. Think about the analogy of a hill and a valley and the potential energy $V = mgh$, where $m$ is the mass of a ball you are holding, $g$ is the acceleration due to gravity, and $h$ is some height. Clearly we need to choose where to measure the height from, and this is equivalent to choosing the 0 of the energy in the previous example. Let's choose the height to be measured from the top of the hill. Then the gravitation potential energy at the bottom of the hill is negative. What happens to the ball? It rolls down the hill. This is the same with electrons and protons, they see a lower energy if they combine together.
Yes, photons carry positive energy, and can undo the bond. The energy of a photon is proportional to its frequency, and frequencies are always positive. This like kicking the ball when it's sitting at the bottom of the valley. If you kick the ball too weakly, it will just roll back and won't go over the hill. If you kick it strong enough, it will get through, and might have leftover energy to fly in the air afterwards. Same thing with hydrogen atom. If the photon has more than 13.7 eV of energy, it will separate the atom, and the electrons and proton will have some kinetic energy (they will move with some speed). If it is lower than that, than the electron and the proton will get farther away, but will not separate entirely.
Hope this helps,
Stefano
