While reading Richard II, I came across the word unpossible:
BUSHY: For us to levy power proportionable to the enemy is all unpossible.
This is the only use of unpossible in all of Shakespeare's plays. Impossible is used 44 times. I don't think the answer to "Why did Shakespeare only use unpossible once?" can be answered "Who knows? It's Shakespeare, anything goes." The use of unpossible is also in the 1597 quarto.
I also had difficulty finding online information about unpossible, other than that it is archaic and not used. I recently saw a game listed on Itunes and Googleplay called Unpossible.
When did the change to impossible happen? Why did it happen? There are many words in English that use the "un" prefix to say "not something:" Unjust, unfair, unloved, unhealthy, etc. When did unpossible stop being considered useable?