I am at an loose end with one particular box that is running SQL Server 2008 R2 Express.
I recently encountered the same issue however I am using Windows 10 Technical Preview Build 10041 and SQL Server 2014 (Advanced).
Follow the same advice from above:
In particular, my problem was that I did not enable the TCP/IP in Sql Server Configuration Manager->SQL Server Network Configuration->Protocols for SQLEXPRESS.
Once you open it, you have to go to the IP Addresses tab and for me, changing IPAll to TCP port 1433 and deleting the TCP Dynamic Ports value worked.
Follow the other steps to make sure 1433 is listening (Use netstat -an to make sure 0.0.0.0:1433 is LISTENING.), and that you can telnet to the port from the client machine.
Finally, I second the suggestion to remove the \SQLEXPRESS from the connection.
AND ----> That last line is important! It looks like to be Windows 10 specific; remove \SQLEXPRESS from your connection string. What was frusting was that SQL Management Studio connects just fine using either connection string (full or short), however Visual Studio only accepted the connection string without the \SQLEXPRESS.