//using our client context, load the folder of the files we want links to
Folder folder = Program.spConnection.context.Web.GetFolderByServerRelativeUrl(Properties.Settings.Default.SPImages);

FileCollection filec = folder.Files;
Program.spConnection.context.Load(filec);
Program.spConnection.context.ExecuteQuery();

List<string> allSharedURLs = new List<string>();
//Load Each File in the folder
foreach (File f in filec)
{
    Program.spConnection.context.Load(f);
    Program.spConnection.context.ExecuteQuery();

    // I would like to add the publically shared url to the list at this point
    // allSharedURLs.Add( theLink );
}

var item = //code to get list item..
var objSharingInfo = ObjectSharingInformation.GetObjectSharingInformation(
        clientContext, item, false, true, false, true, true, true, true);
clientContext.Load(sharingInfo);
clientContext.ExecuteQuery();
//objSharingInfo.AnonymousEditLink
//objSharingInfo.AnonymousViewLink