Kamran,
Facebook login (doc) supports OAuth workflow so you should be able to follow the same implementation for ArcGIS Online (also OAuth) login that we have in our templates (including MapViewer) and our sample OAuth + Biometric Authentication.
You would need to get the required parameters from your Facebook developers account, for example, client_id, reponse type, valid OAuth redirect URI, etc.
As for sharing the Facebook's location to MapViewer, it should be doable, as long as you know how to get that location using Facebook API, and it returns some sorts of location data (lat/long, x/y, address, etc), we should be able to take it (convert it if necessary), then display it in MapViewer and do further GIS related operations.
I hope this helps,
Erwin