Critical mass point moving (changing the course of a Star) is impossible unless you are a God (not Q or some other lame intangible wimp)
It can't be done

The only way to Steer a star would be to move its centre of gravity (by several hundred thousand miles Minimum)
This is utterly impossible for any being less powerful than a God

Actually, if you applied enough force to the star to accelerate it along a trajectory, whilst applying enough force for an equal acceleration on the actual sphere, the star will remain at the center of the moving sphere. The problem here would be in either A: applying said force to either the star or the sphere or in B: coordinating its application perfectly - one newton in the wrong direction at the wrong time, and your star is no longer the center of the sphere.

An alternative would be to somehow connect the star and the sphere via a series of towers extending from the inner surface of the sphere to the closest solid layer of the star (presumably the star's core, though i'm not entirely certain as to the structure of stars), however this would be very unlikely due to the eventual destruction of said towers, unless they were created with special materials (solid neutronium, in the ST example, might suffice). However upon second thought, I find that the gaseous parts of the star would still be unaffected, so it would be a better idea to raise some sort of shield sphere around the star, blocking its movement - presumably it would not block the light and energy of the star, though.

Edit: Did some typo fixing.

*throws Starless Void at a random passerby*
Don't ask but it's about a dyson sphere.

The only way to miniaturize a Dyson Sphere is to have a quantum singularity at the center instead of a star. D'Deridex Romulan Cruisers use artificial singularities in their warp engines. So theoretically this could be possible. A stable quantum singularity would offer unlimited energy if harnessed correctly.

Death Star ftw.

First, I don't know what a "critical mass point is"; it isn't a term used in physics or any astronomical discipline, unless I was sleeping in class that day. But regardless, yes you can move ANYTHING regardless of mass, provided you have enough energy and the engineering technology to put it to use. To move a star artificiallly just takes a lot of mass, a lot of energy (though not nearly as much as you would think), a great deal of time, and a little bit of ingenuity.

You move a star the same way you move an asteroid. To move an asteroid, you start with a ship that is as massive as you can get it and still be able to move it into position. I would place the ship in an eliptical orbit around the asteroid such that with each pass from perigee to apogee the ship gives the asteroid a little tug. Each tiny tug changes the asteroids orbit by a miniscle amount. Enough tugs in the right direction and you can steer the rock almost anywhere you choose as well as increase/decrease its velocity. This will work within a reasonalble timeframe for even a very large asteroid.

The technique can be easily scaled up. If you could steer Jupiter, you could affect the motion of the sun. By steering larger and larger objects, you could very well move Jupiter. Well not YOU as we are talking millenia at least to make this work for Jupiter; even longer for the sun.

And before you say it, this will absolutely work as it is based on leveraging well known and understood principles. In fact, we do something very similar to send probes to distant planets. the only difference is scale.

As for Dyson Spheres, most people completely misunderstood Dyson's proposal. The sphere would not be a solid object; there isn't enough matter in the entire solar system to build something like that, and if you have to travel to another star for supplies, what's the point? A Dyson sphere consists of an interconnected (not necessarily physically connected) web of structures designed to collect nearly all of a star's energy. Like most things ST, the science is riidiculous, but based on some grain of real science. You can read all about real Dyson spheres here. When I was a kid, I even desgined my own version after reading Ringworld, so I know a little about the subject.

Who said anything about miniaturization of dyson spheres?

No physical object that we can make can get within "push" distance of a star without being destroyed and a star is connected to other stars by its place in the cosmic dance (gravity well attractions , orbits and the whole revolving galaxy)
to move it you need to change its centre of gravity dramatically (and use more energy than a Sun produces)
so you can't

practically NO... mathematically, YES. For instance, if you put a mass in proximity to a star, they will start to move toward each other. Those things exist, twin star systems for instance.

I'm not sure about black holes though. my guess would be yes. thoughts?

Feydman and some others explained how it could be done.
very good and simple explanations.

If you mean moving black holes, the answer is yes. If I am not mistaken, each known galaxy contains at least one black hole at the center. And the galaxies are quite obviously moving, otherwise astronomers wouldn't predict a collision between Andromeda and the Milky Way soon(tm). (See what I did there? :p)