What object could I have seen at late June on the Sky? In around late June, I saw two star-like bodies in the sky. At first they appeared to be twinkling, but when I looked closely they appeared to change color; blue, green, a slight red, then yellow.

It freaked me out.

But I'm wondering, did I see dying stars or was it some sort of rare phenomenon occurring because of the earth's atmosphere or was it something else?
Checked on Sky Map on my phone for late June, 2016 in Kolkata - and Saturn and Antares were close to each other around then in the Eastern sky, so these may be the two objects you saw.  The colour change sounds like atmospheric effects affecting the seeing that night.  I don't know if June in Kolkata is particularly worse than usual for atmospheric conditions, but even on a seemingly clear night there can be a layer of turbulent air above you that will diffract the light from the sources and produce unsteady images.  This is usually worse closer to the horizon as well (as these two objects were).
Planets don't usually twinkle, so if Saturn was one of the objects, then it must have been a strong effect.
Nothing to freak out about, and keep looking up! :)