I did the following test:
1) I already have ArcGIS for Server 10.2.1 and SQL Server Express 2008 R2 running on our Application Server Machine. On this same machine are 4 SQL Server Enterprise Geodatabases which are running no problems.
2) I created a 5th SQL Server Database (DBO Schema), then was able to successfully enable enterprise Geodatabase. No problems.
After this test, it appears that the problem is this:
1) The database I want to enable enterprise Geodatabase on resides on a different server machine, that does not have ArcGIS for Server installed on it. However, it does have SQL Server 2008 R2 (full version) running on it. But, with 10.2.1 this should not be an issue.
2) Unfortunately, this server machine is managed along with the SQL Server installed on it by our IT Dept. Whereas the Application Server Machine I mention above is managed by me in our GIS Group (Not in the IT dept, thank goodness). I have no idea what privileges our IT Dept provided me in the SQL Server Database other than what they told me this: "You are using DBO Schema with full editing privileges".
Do you think this problem of not being able to enable enterprise Geodatabase is an issue having to do privileges?