Can a president order the BSA to admit homosexuals by executive order?
Seems doubtful. Every POTUS has served as the BSA's "Honorary President" since BSA's founding in 1910.
On January 13, 2009, BHO was petitioned by dissenters of BSA's membership policy (which forbids open homosexuals and atheists - adult and youth) to renounce his honorific title. He has not done so, to date. He has met with a BSA delegation 4 times now in the White House (once a year - I think always in FEB close to the founding date of BSA) to receive the BSA's annual report which is specified in BSA's national by-laws.
Should BHO be re-elected, I think he will resign as BSA Honorary President. As the title implies, he has no real job responsibility in BSA.