Interesting question. I'm reasonably certain that the answer boils down to a property of matter called the equation of state.
An equation of state is a relation between several thermodynamic variables, typically pressure ($P$), density ($\rho$), and temperature ($T$). You might have heard of some simple ones, such as the ideal gas law. If we know the equation of state of a substance, we can figure out how it will behave under certain circumstances.
The matter in a normal main sequence star can be described, as a good approximation, by the ideal gas law:
$$P\propto\rho T$$
where the constant depends on the composition of the fluid.
This equation of state to a nice effect informally called the "thermostat mechanism", which stabilizes the star against instabilities. If the temperature rises due to higher fusion rates or some other perturbation, so will the pressure, and the star will expand. The temperature and density will soon decrease, lowering fusion rates. In turn, the pressure decreases, putting the star into a new stable (equilibrium) state. The same happens if the temperature drops.
Degenerate matter does not obey the ideal gas law. In objects mainly supported by electron degeneracy pressure, there is a different equation of state. The non-relativistic form is:
$$P\propto\rho^{5/3}$$
while the relativistic form is
$$P\propto\rho^{4/3}$$
Note the total lack of temperature-dependence. This means that when the temperature rises in a body supported by this pressure, there is nothing akin to the thermostat mechanism to lower the temperature. The results can be catastrophic. Type Ia supernovae, for instance, happen when a white dwarf accretes matter and begins fusing it on its surface. If the fusion rate is too high, the temperature rises, and runaway fusion may destroy the star.1
A sort of dance between the two relations is the mechanism behind a helium flash. In a star slightly more massive than the Sun ($\sim2$ solar masses), when hydrogen fusion ends in the core, the star shrinks a little, as fusion continues in the outer layers. Eventually, degenerate matter is formed in the core, and the inner part of the star is now supported by degeneracy pressure, rather than thermal pressure. Eventually, the core shrinks enough and gets so hot that helium quickly fuses into carbon, creating a seemingly runaway reaction. However, as the star expands again, thermal pressure in the outer layers takes over, calming the star. A large amount of energy is released (hence the name "helium flash"), but the star survives.
Your scenario is interesting because if the other side of the wormhole opens into any high-temperature, high-pressure environment, there will be absolutely disastrous consequences. The only hope for there not being an explosion would be if, for some reason, the wormholes are extremely short-lived most of the time, and so only a very small amount of material can travel through.
The semi-handwavy explanation you might be after is that the wormholes form because of some temperature differential. The Bose-Einstein condensate is cold, and the star-like object is hot. We could pretend that only high differences in temperature allow the wormholes to exist at all.
Also, assume when the wormholes are opened into whatever target object you're after, cause increases in temperature. At first, this isn't a problem, because the amount of matter the scientists are using isn't that large. After enough time, however, this effect builds up if it is not mitigated by the target body through cooling (which you could claim then closes the wormholes, allowing only a small amount of matter to go through), and an explosion occurs.
If you want to do something like this, then I would recommend using a white dwarf in a binary system that is already accreting matter. The wormholes simply provide the catalyst for fusion to begin, and there is no way to stop it. Higher temperatures mean a higher temperature difference. Thus, the wormholes stay open longer, and all of a sudden a significant amount of matter travels through - as the white dwarf undergoes a Type Ia supernova.
Side note: Neutron stars
One thing I realized I totally ignored was the equation of state for neutron stars. There are two reasons for this:
- It must be relativistic, meaning it's quite complicated.
- It's not actually well-known, and is an active area of study.
Neutron stars are massive and compact, and therefore an accurate treatment of them needs to use general relativity2 (although non-relativistic approximations could work to some degree; if you're really curious, you could look at polytropic models with indices of $0\leq n\leq1.5$). The other issue is that we don't yet know what equation of state best describes neutron stars. More observations (especially of binary systems and pulsars) are needed to place appropriate constraints, although several have already been proposed. Lattimer (2013) is a good review of various neutron star equations of state.
For this reason, I've decided not to discuss this option; we simply don't know enough to determine exactly what the effects would be in your scenario. I can tell you, though, that they would not be good for the physicists.
Other degenerate objects, like quark stars, also have complicated and not-yet-well-constrained equations of state. I talk about these in some more detail in another answer of mine.
1 This isn't guaranteed to happen; the shockwave may instead fizzle out, in what is called a deflagration wave (as opposed to a successful detonation wave). Modeling how these types of waves arise has been the target of simulations for quite some time.
2 Relativistic models are also quite nice for white dwarfs; see, for instance, this project.
