Do all stars have the potential to have life supporting planets?

Great question. The goldilocks zone is usually defined in terms of a region where the equilibrium temperature of the planet lies between some temperature limits (these temperature limits are somewhat debatable, but irrelevant for the purposes of this question - the boundary becomes fuzzy). This region can be calculated by working out how much flux is received from the star at a given radius.

In that sense - all stars have a goldilocks zone - you can always work out some distance from a star where the flux is roughly equivalent to the flux we receive from the Sun for instance. It is much closer to a dim star and much further away from a luminous star.

But, there is a difference between a habitable zone and a continuously habitable zone. If you seriously want life to evolve on a planet, then it will take time. How much time? No-one is sure but it seems to have taken a few hundred million years in our solar system. There are also good reasons to suppose that young planets are not going to be habitable - either they are still extremely hot after formation or they are being bombard by debris.

Thus you can probably exclude any star with a main sequence lifetime of less than about 100 million years (or longer if you feel conservative about how long it takes life to get going). This rules out stars of more than about 5 solar masses.

You might also rule out stars that evolve quickly. If a star changes its luminosity quickly on a short timescale then the habitable zone moves drastically too. This will happen near the ends of the lives of all main sequence stars and for all subgiant and giant stars. So for these, yes there is an instantaneous habitable zone, but no region is in a habitable zone for hundreds of millions of years or more.

You might want to consider not only the luminosity of the star, but its spectrum as well. For instance, both hot white dwarfs and low-mass M-dwarfs are reasonably stable and long-lived; they are faint, so have habitable zones close to the star (much closer than 1 au). However, for different reasons, both these types of object emit copious UV radiation. In hot white dwarfs it would be because they have hot photospheres. In M-dwarfs they can be highly magnetically active into old age, having a hot chromosphere and corona that would strongly irradiate any nearby planet. You might consider that their "habitable zones" were in fact uninhabitable. Of course an atmosphere and a strong magnetic field might counteract this.

A lot of this depends on your terms of reference and definition. The wikipedia page on habitable zones has quite a nice discussion, which emphasizes the debatable nature of this topic.

A final thought, which is not in the wiki page. Multiplicity could mess things up. Planetary systems will not be stable in the habitable zone of either star in a binary system if their separation is comparable with the orbital radius of the habitable zone. If you had two stars like the Sun, their separations would have to be less than a few tenths of an au or greater than of order 10 au in order to allow something to orbit at about 1 au around one of them. Even then, there could be all sorts of dynamical instabilities which prevent long-lived planets, especially if there were other planets in the system too.