What kind of fabric is used for the Eames fibreglass shell upholstery? Dye experience?  

I don't know anything about removing the fabric but I think the fiber content is nylon. I'm not sure. I am pretty sure it's synthetic of some sort.

Here's how to determine fiber content by burning:
http://images.taunton.com/downloads/th/T168_Burn_Test_Chart.pdf
All you need is a very small bit of fiber. Maybe it's possible to snip a bit from the edge? Use tweezers to hold it over a candle flame. Don't use a match because the smell of the burning match material will interfere with the smell of the burning fiber, which is one of the clues in identifying it.
You can read about dyeing nylon and other fibers on this site:
http://www.dharmatrading.com/dyes/dyes-for-dyeing-silk-wool-nylon.html
I don't know if you can find those dyes locally; if not, this place probably will ship to you. They are pretty wonderful! I've bought lots of dyes and supplies from them in the past.

slightly stained vintage orange is better than bad newly dyed black
you will not be able to put it back together if you take it apart
if you use the chair regularly your own slime will build up on it anyway so I say don't sweat it

thank you!
appendix: the Hopsak is 100% Polyamid Nylon
i gues i will try a dye. hope i can report something positiv