I can tell you that you need banner paper or you'll be investing in a lot of Scotch tape. I don't know about EQ but you should be able to use Corel Draw. Sorry I can't help more.
I do have a question-why do you want to draw your own? In the long run, I don't know if you'll save much money. I figure there are so many great designers out there, I'll keep doing what I do best-quilting client quilts.