Currently our login screen only shows the GoogleApps login button, and the fields for entering a My Atlassian account. I would like to not require external people to create a My Atlassian account just to log into our ticket system.
To clear that up, the login info on your page isn't for My Atlassian, it's for your specific JIRA instance. Unfortunately it's not possible to remove these login fields from that screen. If a user does not have a Google account on your domain, they need to have an account within your OnDemand instance in order to login. Alternatively, you can allow users to create tickets anonymously via the link that Ricardo posted.
I believe this documentation might help: https://confluence.atlassian.com/display/AOD/Managing+anonymous+access#Managinganonymousaccess-EnablinganonymousaccessinJIRA
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