Kip Thorne's hoop conjecture states that things will become a black hole when a hoop of circumference $2\pi R_s$ can be placed around the thing and rotated ($R_s$ is the Schwarzschild radius of a black hole with the mass of the object). In the picture above the hoop conjecture would immediately predict that the black holes would merge around the star - the radius of a black hole with the mass of 8 about equal black holes is 8 times larger.
In this case the result would likely just be a somewhat anticlimatic merger since light and energy from the star being affected would never get a chance to get out of the new big black hole. Event horizons are horizons and can spread faster than light, so basically what you see in this case is 8 holes approaching the star and just as they get really close they turn into a wildly wobbling huge black hole. A bit like 8 mercury droplets joining together into one big droplet.
Note that these are very heavy black holes. We know this because in the picture they have a radius comparable to the star. Sure, a M star is small compared to the Sun (a M9 dwarf is about 8%, 55,656 km), but the Schwarzschild radius of a sun-mass black hole is a bit over a kilometer. So these have to be supermassive holes with tens of thousands of solar masses. Incidentally, this makes the tidal forces weaker: small black holes have a much steeper gravity field so the difference in attraction between the feet and head of someone falling in will be so large they get "spaghettified".
Now, if you have a ring of $N$ mass $M$ black holes they will, if the hoop conjecture is right, merge when the ring radius is about $r=2GM(N-1)/c^2$ (why? Because the hoop has to have circumference $2\pi(r+2GM/c^2)=2GMN/c^2$). To get them to merge at the star radius $M(N-1)$ must be $3.7\times 10^{34}$ kg. If we use $N=8$ then the mass would be 2691 solar masses each. If we use a ring of $N=1000$ they only need to have 18 solar masses of mass, and for 200,000 they would each just have the mass of the central star.
But the implosion would still be boring since at the point where they coalesced there would not be much reaction from the star matter - the holes would be approaching at lightspeed, and then suddenly an event horizon would form engulfing the star.
To get some drama, we can consider a ring that turns into a black hole inside the star. In this case a lot of star matter will suddenly discover than it has a lot of potential energy and fall inward. If the new hole is big it will just quietly pass through the event horizon and disappear - some energetic radiation is released, but not much since it is also very redshifted. A fraction of the mass of all of the black holes (many times the mass of the star!) will be released as gravitational waves that will churn things up a bit, but it is surprisingly mild (partially because of the very symmetric implosion; higher order multipole moments decay fast).
If the new hole is small the star has to compress a lot to get in. This will heat up the plasma, and this creates a counteracting pressure. So now things turn very energetic. Had the star rotated fast an accretion disk would have formed as plasma swirled around, releasing its potential energy before finally falling into the hole at the center. This can be as bright as an entire galaxy. However, here it is more of a spherical ball of super-hot plasma with slightly less hot plasma falling onto it. I think the best description is something very much like a supernova. After much of the star has blown away remaining gas will form a hot accretion disk around the black hole.
To sum up, yes, the star will be compressed. But not so much by the ring, more by the internal black hole.
