StringCollection is a specific type. You cannot convert from List<t> to it. However, you don't really have to use StringCollection. That's just in the designer.
You have two choices
(1) Save your strings to StringCollection instead
(2) Change the auto-generated code in Settings.Designer.cs
I recommend the first, because changing auto-generated code can be problematic. The one issue I know is that you'll need to repeat the change anytime you add a setting through the designer.
The second option is simple to implement. Open the Settings.Designer.cs file and change the System.Collections.Specialized.StringCollection to System.Collections.Generic.List<string>. That's it, at that point you can write your normal code to manipulate the setting (it's not like you're going to use the designer anyway for the value).
if (Settings.Default.TestSetting == null) {
Settings.Default.TestSetting = myValues;
Settings.Default.Save();
}