Have you checked the shared memory in SQL Server protocols? If it is disabled, you will not be able to connect to SQL Server in Windows 10.
On contrary, in Windows 7, 8 and 8.1 enabling the shared memory was often causing the same effect, so it is by the rule that it is often disabled. After upgrading to Windows 10 you might hit the issue as now it is required to be enabled... not sure why MSFT made this mess, but I have seen this issue quite often after upgrading to Windows 10
Run SQL Server Configuration Manager and then navigate to SQL Server Native Client Configuration
Check Shared Memory and if disabled, enable it
Try to connect SQL Server again