And the answer to that is almost certainly 'no'. The Solicitor General speaks for the United States in cases that appear before the Supreme Court. Not district courts or appeals courts, but the Supreme Court. Not a single one of the Birther cases that have been submitted to the Supreme Court to date has passed the conference stage. None have been heard by the court, so none of them have involved the Solicitor General.
The Solicitor General is often, but not always, involved before the Supreme Court agrees to hear a case-- the Solicitor General sometimes files a brief asking the Court to grant or deny cert. (Hence Kagan's many recusals.) But the Solicitor General never filed anything in any Obama eligibility case.