Ok, now I want you to understand why that is. The "\\" is an artifact of the C# language, not connection strings. Sql Server only uses one backslash between server name and instance. "\" begins an escape sequence in C#. "\n" is newline, "\t" is tab, etc...So, since "\" is a sequence, how do you just render a single backslash? By using two. "\\" renders to "\".
Of course, that's only for strings in your code. Any string from any external resource, be it the console, a text file, network communication...they don't have escape characters. They've already been rendered.
That's why you only need one backslash in your text file.