Can there be eternal stars? the question is quite straightforward:


*

*Can there be stars that shine forever without ever collapsing nor growing?

*Do we know some really, really old stars? (whatever age that might be)
I hope to get answers from physicists, as for the nuclear reaction constraints involved; but I'm also looking for the point of view of cosmologists and astrophysicists.
 A: The word thing you are looking for is "Black Dwarf" stars. Which are White Dwarf stars which have cooled to match the temperature of the cosmic background. Since this is likely to take more than the current age of the universe, there aren't any. These will exist forever, unless hypothetical proton decay finishes them off or a hypothetical Big Rip due to Dark Energy does so.
Shining forever isn't going to happen because mass into energy would still destroy them no matter what mechanism existed
A: (By stars I'm assuming you're implying stars like the Sun, which are a majority of the stars we see. @Dirk Bruere's answer about Black Dwarves is correct. ) 
No, I don't think they can. The primary process that 'fuels' stars is nuclear fusion. In the process of nuclear fusion, lighter elements fuse together, releasing a tremendous amount of energy (because the fusion product has greater binding energy than the reactants.) This, however, is only true until elements around Iron. Once Iron is obtained as a fusion product, it is not favorable to fuse Iron and its neighbors into heavier elements because the products obtained have lesser binding energy.
 
Stars exist in their observable state because of the intricate balance between the outward flux of radiation and particles from fusion, and the stars own gravity. Our sun, for instance, is in its Main Sequence stage, and is currently shining because of Hydrogen fusion. When the hydrogen in its core becomes significantly less, the drop in radiation pressure would cause it to collapse and constrict its core, which would cause helium to fuse. This cycle of expansion and contraction can only continue until certain elements, because some stars are just not heavy enough to provide energy for the fusion of heavier elements. (For example, the sun's fusion process would be favorable only till elements close to Carbon.)
For lower mass stars, when the end point of this cycle is reached, the outer layers of the star are shed off while the core stabilizes.
Core collapse takes places for very massive stars, because of reasons I mentioned in the first paragraph. (This, however is not the end of a star's life cycle. Read this: https://en.wikipedia.org/wiki/Stellar_evolution)
Summarizing, stars (in their shining form) can't live forever.
A: Answer from astrophysicist.
Point 1.
The problem with eternal shining is that star loses energy with photons (and some material) and it'll need income of energy from somewhere.
There are brown dwarfs, black dwarfs, black holes etc., which are just remaining of stars and don't 'shine' (or, in case of white dwarfs, fade out to the point we cannot detect them).
If you replace word 'forever' with 'REALLY long time', the it would be low-mass stars (approx. 0.07-2 solar masses), especially red dwarfs (theoretically).
Point 2.
The oldest stars should be so-called population III stars, which were first to form from primordial H, He and some Li.
As age of the Universe is larger than evolution time for most stars, it's not that easy to find one. Though there is some evidence published recently (from a long time ago in a galaxy far, far away...)
