I have been struggling with this for days and have not found a solution. First I had problems even connecting the the SQLexpress instance but figured out that it was the firewall preventing me from accessing from another computer. I turned off the firewall and I can connect to SQLExpress and create a geodatabase but as soon as I try to create a new feature dataset, I get "Cannot connect to Database. Bad Login User" If I disconnect and reconnect, that geodatabase also disappears.
I have reinstalled sqlexpress about 20 different times trying different things to no avail. Anyone have any suggestions?