Congressman Brendan Boyle (D-Pennsylvania) has introduced a bill to require that the Speaker of the House be an elected member of the House. The bill doesn’t have a bill number yet. Article II, section two of the U.S. Constitution says “The House of Representatives shall chuse their Speaker and other Offices” but does not say the House must choose a member. However, all speakers in history have been members of the House.
The presidential succession act says that if the president and vice-president both die, the Speaker shall become president.