Yup the Collector app supports token, basic, digest secured services. The maps require there be an editable feature service to be present and if they were created a while ago you may need to resave them to pick up the Collector flag. If you are using internal secured services editing can be disabled in the web map which can restrict the Collector flag from being written out. You could work around this by adding in an editable hosted feature service like the location tracking layer and resaving the map. Then you can pull to refresh your All Maps section of the app and you should see this new Collector ready map.