I wonder whether the nuclear fusion of brown dwarfs and more massive stars really depends on their mass only or if it could also start nuclear fusion if it is dense enough but not as massive as brown dwarfs usually are (they need more than 12 Jupiter masses to ignite deuterium). It is like with a nuclear fusion bomb. Since hydrogen bombs explode because of their density and not only their mass, can this also happen to stars? Or did I miss something? Or is a density to ignite deuterium at lower masses simply impossible? If I'm right, what would be the critical density to ignite deuterium at 1 Jupiter mass?
Or did I miss something about hydrogen bombs?
You missed something about hydrogen bombs.
The center of our Sun is the equivalent of a warm compost pile in terms of energy produced per unit volume. A multistage thermonuclear weapon briefly (very briefly) compresses and heats the fusible material in the bomb to conditions far beyond those found in our Sun, let alone Jupiter.
Suppose you had a giant planet with a central temperature/density too low to sustain D fusion (i.e. below about 13 Jupiter masses). You then magically are able to increase the density by somehow driving the mass of the planet inwards (which will actually increase both the density and temperature).
It is the temperature rise that is important. The energy per unit volume generated by fusion is $\propto \rho T^n$, where $n\gg 1$.
All that would happen is that the fusion rate would become significant when $T\sim 3\times 10^5$ K, but the object is now completely out of equilibrium due to its higher central pressure. It will expand and the fusion reactions will be damped down.
The key to making an explosion happen is to get all the energy released from the fuel before that energy can be transported away from the burning site. In the case of planets/brown dwarfs, both conduction and convection are rather efficient energy transport mechanisms. Even so, there is a significant increase in luminosity and a $\sim 30\%$ increase in radius when D is ignited in a planetary mass object that accretes enough material to ignite D at its core (Bodenheimer et al. 2013). But this only happens because the weight of the object is sufficient to sustain a relatively low rate of D fusion - indeed the reaction proceed over $\sim 100$ million years.
So I think my answer is yes you could ignite D fusion, but to sustain it would mean sustaining whatever mechanism you had used to increase the density/temperature in the first place.
I realise that there is an implicit piece to this argument that I really ought to make explicit. You might have thought that the density of an object could just increase without limit as it cooled and contracted. But no, the reason that there is a lower mass limit for fusion is that these low mass brown dwarfs and giant planets have electron-degenerate interiors. This means the central pressure becomes independent of temperature and that an object below the D-burning threshold can effectively cool at constant density. i.e. Their densities will not increase (much) further and they will simply cool at nearly constant radius. Fusion only occurs if the temperature/density combination reaches a threshold before electron-degeneracy pressure essentially fixes the internal density structure. And this only happens in balls of gas that are greater than 13 times the mass of Jupiter.