I encountered a situation very similar to the one you are encountering. I had to set up accounts for 2 individuals at a school where I am an administrator. Each of these were unique in problem, but was resolved with the same process. The individuals were on brand new iPad 3s on the newest iOS. I didn't want to create new accounts using emails other than their official school emails, which created the issue from the beginning.
Scenario 1: Individual had set up apple id on apple's website after iPad was configured on initial startup without a profile. Every time we tried loggin in with the apple id to make sure it was the one in use on the iPad, it would not give the option for "none" on the payment option. We tried going into the app store both through the app and in the settings menu, and we also tried going through the itunes app with the same results. Just wouldn't work. Why should the user be punished for creating their account after setup as oppsed to before? was my question to apple.
Scenario 2: Individual had not created an apple account when the iPad was set up, but needed to create one to use in the app store. When going through the standard method to create a new account through the app store, itunes, etc, the option "none" is still not available. This one baffled myself as well as the apple technician the most. Apple normally has their stuff in order, but the programmers need to work on these two scenarios.
Resolution: After working as a technician for a University with a few years of experience, I couldn't wrap my mind around a solution until myself and a coworker happened to think about trying to login throuh "iCloud" tab in the "settings" menu on the iPad. We first tried logging in with the individual from scenario 1 to see if it would work with an existing apple id created on apple's website. To our surprise it worked, allowing us to select "none" as a payment option after logging into the appstore with the same id for the first time! We then were hopeful that it may solve the second scenario. We went to create new id under the iCloud login on the second individual's iPad, and created it successfully and was able to choose "none" as the payment option. Apple themselves did not even know how to do this other than suggesting to create new accounts, but I did not see adding a gift card or credit card (personal information) or using a new email (why should I have to?) to solve the problem.
Bottom Line: Hopefully, this helps those that are encountering the same issue, not wanting to create new accounts/give away private information. Please apple, fix this problem in future updates.