I have docker for windows installed on my machine. There is a console application targeting .net core 1.0.0 that tries to access a SQL Server database running on a different VM.
as in this answer
SQL Server instance string connection in Linux Docker
According to Saurabh Singh from Microsoft:
The Instance name support is available in v 1.1 of .Net Core. In v1.0 of .Net Core, Instance names are not supported on OS other than Windows.
So I don't think you can connect from .Net Core 1.0 running on Linux to an SQL Server using instance name.
Your choices seem to be:
don't use instance name wait for .Net Core 1.1 (planned for "Fall 2016") use pre-release version of .Net Core 1.1