First and foremost, the BEC systems studied in detail today do not involve the formation of any bonds between atoms. Bose-Einstein Condensation is a quantum statistical phenomenon, and would happen even with noninteracting particles (though as a technical matter, that's impossible to arrange, but you can make a condensate and then manipulate the interactions so they are effectively non-interacting, and the particles remain a condensate).
The "high school physics" version of what happens at the BEC transition is this: particles with integer intrinsic spin angular momentum are "bosons," and many of them can occupy the same energy state. This is in contrast to particles with half-integer spin, such as electrons, termed "fermions," which are unable to be in exactly the same quantum state (this feature of electrons accounts for all of chemistry, so it's a Good Thing). When we talk about a confined gas of atoms, quantum mechanics tells us that we must describe it in terms of discrete energy states, spaced by a characteristic energy depending on the details of the confinement. Because of this, the two classes of particles have very different behaviors in large numbers.
The lowest-energy state for a gas of fermions is determined by the number of particles in the gas-- each additional particle fills up whatever energy state it ends up in, so the last particle added goes in at a much higher energy than the first particle added. For this reason, the electrons inside a piece of metal have energies comparable to the hot gas in the Sun, because there are so many of them that the last electron in ends up moving very rapidly indeed.
The lowest-energy state for a gas of bosons, on the other hand, is just the lowest-energy state available to them in whatever system is confining them. All of the bosons in the gas can happily pile into a single quantum state, leaving you with a very low energy.
It turns out that, as you cool a gas of bosons, you will eventually reach a point where the gas suddenly "condenses" into a state with nearly all of the particles occupying a single state, generally the lowest-energy available state. This happens with material particles because the wave-like character of the bosons becomes more and more pronounced as you lower the temperature. The wavelength associated with them, which at room temperature is many times smaller than the radius of the electron orbits eventually becomes comparable to the spacing between particles in the gas. When this happens, the waves associated with the different particles start to overlap, and at some point, the system "realizes" that the lowest-energy state would be for all the particles to occupy a single energy level, triggering the abrupt transition to a BEC.
This transition is a purely quantum effect, though, and has nothing to do with chemical bonding. In fact, strictly speaking, the dilute alkali metal vapors that are the workhorse system for most BEC experiments are actually a metastable state-- at the temperatures of these vapors, a denser gas would be a solid. They form a BEC, though, because the density of these gases is something like a million times less than the density of air. The atoms are too dilute to solidify, but dense enough to sense each others' presence and move into the same energy state.
The underlying physics is described in detail in most statistical mechanics texts, though it's often dealt with very briefly and in an abstract way. There are decent and readable descriptions of the underlying physics in The New Physics for the Twenty-first Century edited by Gordon Fraser, particularly the pieces by Bill Phillips and Chris Foot, and Subir Sachdev.