It is quite easy to get the "Everyone" account name, with the help of the DirectorySecurity and SecurityIdentifier classes. Here is a small example of getting the name and allowing the user to save a file inside a location with the correct permissions:
string PATH = "";
if (!Directory.Exists(Path.GetFullPath(PATH)))
{
Directory.CreateDirectory(PATH);
DirectorySecurity dirSec = Directory.GetAccessControl(PATH);
SecurityIdentifier siEveryobody = new SecurityIdentifier(WellKnownSidType.WorldSid, null);
dirSec.AddAccessRule(new FileSystemAccessRule(siEveryobody, FileSystemRights.Modify | FileSystemRights.Synchronize, InheritanceFlags.ContainerInherit | InheritanceFlags.ObjectInherit, PropagationFlags.None, AccessControlType.Allow));
Directory.SetAccessControl(PATH, dirSec);
}
using (StreamWriter swWriter = new StreamWriter(PATH, true))
{
sw.WriteLine("TEXT");
sw.Flush();
sw.Close();
}