Build public cast_shell on x86 with GN.

chromecast/common/BUILD.gn

Index: chromecast/common/BUILD.gn
diff --git a/chromecast/common/BUILD.gn b/chromecast/common/BUILD.gn
index 127823b3c0b796dc742ead5f53dc7b218ce74172..9c2ed7b610021700b317b2c100d13477c8bc866a 100644
--- a/chromecast/common/BUILD.gn
+++ b/chromecast/common/BUILD.gn
@@ -2,6 +2,8 @@
 # Use of this source code is governed by a BSD-style license that can be
 # found in the LICENSE file.
 
+import("//chromecast/chromecast.gni")
+
 source_set("common") {
   sources = [
     "cast_content_client.cc",
@@ -14,8 +16,9 @@ source_set("common") {
     "pref_names.h",
   ]
 
-  # TODO(kmackay) Add "platform_client_auth._simple.cc" to sources
-  #               if chromecast_branding != "Chrome"
+  if (chromecast_branding == "public") {
+    sources += [ "platform_client_auth_simple.cc" ]
+  }
 
   deps = [
     "//base",