String.Empty in Switch/case statement generate a compiler error

You can try like this instead:

switch(filter ?? String.Empty)

string.Empty is a read-only field whereas "" is a compile time constant. You can also go through a article here on Code Project String.Empty Internals

//The Empty constant holds the empty string value.
//We need to call the String constructor so that the compiler doesn't
//mark this as a literal.
//Marking this as a literal would mean that it doesn't show up as a field 
//which we can access from native.

public static readonly String Empty = ""; 

On a side note:

You will also see this issue when you are providing the default parameter value inside your method(C# 4.0):

void myMethod(string filter = string.Empty){}

The above will result in a compile time error as the default value needs to be a constant.

The reason is: you cannot use readonly values in case: consider the following scenario:

public string MyProperty { get; } // is a read-only property of my class
switch (filter)
{
    case MyProperty:  // wont compile this since it is read only
    break;
          // rest of statements in Switch
}

As you said string.Empty is equivalent to "", here I can prove this with the same example of a switch statement:

string filter = string.Empty;
switch (filter)
{
   case "":  // It's Okay.
   break;
    //rest of  statements in Switch
}

Then the only reason it won't allow string.Empty in case it is read-only, the switch won't allow read-only values in its case.