What you describe looks like a border case for a Dyson sphere. Its effects would depend on what the sphere is capable of.
I think that nothing much would happen (but I'm definitely not an astrophysicist). A star is a large fusion explosion happening, and it exists in hydrostatic equilibrium between its gravitational force (that tends to pull it inwards) and the thermal energy release (that tends to have it balloon into a nebula).
Reflecting back the whole radiative output of the star would lead to a negligible rise in temperature (where it counts, temperature is on the order of millions of K), and the star would proceed slightly along the main sequence - it would "age" a little faster, and probably its evolution would be skewed, so that in a couple billion years a borderline yellow dwarf might behave like an orange dwarf. Also, incredible as it may seem, a star is actually quite opaque and already not as good a thermal conductor as you might think, so upping the opacity to 100% isn't so radical a change.
Actually, something sort of like that already happens in some stars - the so-called Cepheid variables. In those stars, mass and temperature combine in such a way that a whole layer of the star finds itself in a condition with some properties of your handwavium sphere; namely, its radiative opacity is markedly lower than normal, and some energy is reflected back into the core. This makes the star age faster and burn hotter, but the raised temperature converts the handwavium layer back into transparency; the extra energy is radiated away and the star appears more luminous for a while. Then, the energy loss cools the star a bit, and the handwavium layer re-forms, and the cycle begins anew. This is called the Kappa mechanism.
Partially reflecting the output (e.g. leaving an exhaust nozzle, or using a Dyson semisphere) would result in thrust being applied to the whole assembly.
Simple reflection, with unstable star
If the star were exactly in the right and unlikely condition - a hot, blue star with recent massive low-metallicity mass influx (typically from a binary companion or passer-by star), which by the way would be probably enough to trigger a star explosion - then the handwavium sphere could induce a runaway fusion process in a much-larger-than-normal volume of the star. Normally the star would get rid of the excess energy by massive flaring or further expanding, followed by cooling, and all you would get is a planetary nebula. This might be, roughly, what is now happening with the star Eta Carinae.
Reflection and containment
If the handwavium is able to sustain the pressure increase as well as the radiation, then the star might undergo photodisintegration, or the instability might reflect inwards, lead to a core collapse, and we get to see whether the handwavium can withstand a supernova explosion. If it can't, the braking effect should be enough to trasform it into a hypernova explosion (roughly the same bang, but much more luminous).
Reflection and indestructible containment
Otherwise, a supernova-proof reflecting enclosure would be a sure-fire method to guarantee that any star above the Landau limit - we should probably factor in some higher-than-normal neutrino loss - would ultimately collapse into a black hole. It couldn't happen to the Sun, since this limit is about 1.5 solar masses. "Ultimately", because the time to do so could well be of the order of magnitude of a normal star lifespan, especially if the star isn't too big and energetic to start with.
(It has been pointed out to me that such an enclosure would already be a black hole unless examined from planetary distances. For it would be a zone from which "nothing can exit, not even light", and yet there would be a gravitational field associated with the mass of the enclosed star).
Life outside the edge
A semi-permeable or locally-permeable handwavium enclosure with the appropriate radius (to ensure a suitable surface gravity) would also be habitable (a true Dyson sphere). The energy sources would be "wells" drilled in the enclosure, which could then spew out plasma at a temperature of several tens of thousands of K. Using high-altitude passive radiators with a temperature below 300 K, this would yield a thermal efficiency in excess of 97-99% with almost no other technology - a simple heat engine made of handwavium would do.