What are the minimum element requirements for a star? What elements would be essential for a star to exist? 

From what I understand our sun is made of mostly hydrogen and helium and most stars have small amounts of heavier elements like carbon, nitrogen, oxygen and iron. 

But could a sun exist on a single element such as Hydrogen? Or on another gas for example oxygen or nitrogen? Or even rare gases such as Neon or Xenon?

Is it even possible for a star to exist without any hydrogen or helium?
Yes, a star can exist that's entirely hydrogen. Hydrogen is the fuel that makes stars happen unless they're very large. A star made entirely of hydrogen, so long as it was massive enough, would be very similar to the stars we see. The "metalicity" which refers to non hydrogen-helium elements has some effect on the star's rate of fusion, density and spectroscopy but it doesn't have a major effect.

When the universe was very young and almost entirely hydrogen and helium in terms of regular matter or elements, star formation was difficult because those light gasses were difficult to clump together.   The first stars were probably very large as a result, but once heavier elements began to be released in early supernovas, then star formation became easier because there were elements that could clump together and help smaller stars form.   So the other elements play a key role in star formation, but a much smaller role in the life of the star.

Very large stars fuse heavier elements, but as I understand it, fusion of heavier elements is more explosive. Hydrogen is wonderful in the way that it's very unlikely to fuse, so even large stars take millions of years to fuse their hydrogen because something like 99.9999% of the time, when two hydrogens get close enough to fuse, they just return to being two hydrogens, and only 1 in a hundred million billion billion times (1 in 10^26), credit to @PM2Ring.  It's only very rarely that the proton-proton collusion undergoes the fusion process and becomes a deuterium.

This very slow process doesn't happen with the heavier elements, so a star made entirely out of elements heavier than hydrogen and helium would begin it's life as a kind of white dwarf star, that is, the mass of a star but the size of a planet, and technically, it wouldn't be a star at all because it wouldn't be undergoing fusion and when the fusion process finally starts, it would be more like a type 1A supernova, happening very quickly and explosively.  

The CNO process which creates some of the heavier elements in larger stars doesn't happen without hydrogen either, or, more accurately, free protons.

Basically, you can make stars out of pure hydrogen. Other elements would be made inside stars, but you can't make stars out of any other elements, unless you consider variations of a type-1a supernova a "star".  I personally don't.