No. Snowball is not the one who has a dream that shows the animals living happily without human beings around. The pig that has that dream is called Old Major. Old Major has the dream, but then dies before the revolution can happen.
After the revolution, Snowball becomes important. He and another pig named Napoleon are the leaders of the revolution and of the government that is created after the animals get rid of Farmer Jones. But Snowball is not the one with the original dream -- he is one of the main pigs who tries to make the dream into reality.