Here’s an attempted explanation

Quinones are produced by epidermal cells for tanning the cuticle. This exists commonly in arthropods. [Dettner, 1987]

Some of the quinones don't get used up, but sit on the epidermis, making the arthropod distasteful. (Quinones are used as defensive secretions in a variety of modern arthropods, from beetles to millipedes. [Eisner, 1970])

Small invaginations develop in the epidermis between sclerites (plates of cuticle). By wiggling, the insect can squeeze more quinones onto its surface when they're needed.

The invaginations deepen. Muscles are moved around slightly, allowing them to help expel the quinones from some of them. (Many ants have glands similar to this near the end of their abdomen. [Holldobler & Wilson, 1990, pp. 233-237])

A couple invaginations (now reservoirs) become so deep that the others are inconsequential by comparison. Those gradually revert to the original epidermis.

In various insects, different defensive chemicals besides quinones appear. (See Eisner, 1970, for a review.) This helps those insects defend against predators which have evolved resistance to quinones. One of the new defensive chemicals is hydroquinone.

Cells that secrete the hydroquinones develop in multiple layers over part of the reservoir, allowing more hydroquinones to be produced. Channels between cells allow hydroquinones from all layers to reach the reservior.

The channels become a duct, specialized for transporting the chemicals. The secretory cells withdraw from the reservoir surface, ultimately becoming a separate organ.

This stage -- secretory glands connected by ducts to reservoirs -- exists in many beetles. The particular configuration of glands and reservoirs that bombardier beetles have is common to the other beetles in their suborder. [Forsyth, 1970]

Muscles adapt which close off the reservior, thus preventing the chemicals from leaking out when they're not needed.

Hydrogen peroxide, which is a common by-product of cellular metabolism, becomes mixed with the hydroquinones. The two react slowly, so a mixture of quinones and hydroquinones get used for defense.

Cells secreting a small amount of catalases and peroxidases appear along the output passage of the reservoir, outside the valve which closes it off from the outside. These ensure that more quinones appear in the defensive secretions. Catalases exist in almost all cells, and peroxidases are also common in plants, animals, and bacteria, so those chemicals needn't be developed from scratch but merely concentrated in one location.

More catalases and peroxidases are produced, so the discharge is warmer and is expelled faster by the oxygen generated by the reaction. The beetle Metrius contractus provides an example of a bombardier beetle which produces a foamy discharge, not jets, from its reaction chambers. The bubbling of the foam produces a fine mist. [Eisner et al., 2000]

The walls of that part of the output passage become firmer, allowing them to better withstand the heat and pressure generated by the reaction.

Still more catalases and peroxidases are produced, and the walls toughen and shape into a reaction chamber. Gradually they become the mechanism of today's bombardier beetles.

The tip of the beetle's abdomen becomes somewhat elongated and more flexible, allowing the beetle to aim its discharge in various directions.