The simple answer: No.
The predominantly brown colouring you see in human eyes is the result of brown pigment molecules in the iris. Changing the Sun's colour won't change that.
The less simple answer: Maybe.
* The colour of an object is just the spectra of light "reflected" by that object. Change the colours available for reflection, you technically change the colour of all objects in that light.
* We don't normally perceive this as being the case because of something known as colour constancy. Our brains work to adjust our subjective experience of colour to give us the illusion of consistent colour under different lighting conditions, rather than different colours with different lighting conditions. Therefore, if we evolved under radically different lighting conditions, it's conceivable that we would perceive colours differently, rather than having brains that worked to adjust colour experience to what humans on Earth consider a normal colour range.
* Blue eyes, unlike brown eyes, aren't coloured from pigmentation. They exhibit structural colour. That might sound a little strange at first, but it starts making sense when you remember light has wave-like properties. Every "colour" actually has a unique wave length, therefore you can use geometry (on the nanometre level) to control what colours go where. In this case, the specific effect in the eye is Rayleigh scattering. Blue light is scattered more than other colours, so we see more of that.
* Under a star with far less blue light than our Sun, you wouldn't see blue eyes as blue. 970
But the consequence of radically different light on blue eyes doesn't end with blue eyes. All eyes are actually blue but with differing amounts of black-brown pigment on top. So blue eyes are actually just eyes with fairly little pigment, and other eye colours humans normally have (green, yellow, etc) are the just the result of different amounts of brown on top of structurally blue irises. If you changed the natural lighting enough to make blue eyes not blue, other lighter eye colours would be affected too. (There would probably be a minimal structural colour effect on the darker colours, not just the dark browns.)
* It's worth noting that while different types of stars do output different spectra of light, the stars that can conceivably host a human-habitable planet put out light that's fairly similar in colour. (The effects would be minimal.) In addition to carefully picking the kind of star you want for your world, think about the atmospheric conditions of the planet in question. You probably want to think about mechanisms to block out certain ranges of colours, since hot things don't usually skip colours. They just skew a bit into the red direction or into the blue direction. It's a detail that won't concern your readers or players very much. In fact, you may not even explicitly explain what colours are screwed with, just what the results are. But it would at least provide a hint of plausibility to people with some physics background. 970