To be honest, that is a minefield. Imagine if the Military refused to take orders because Bush was “Selected not Elected”.
However ..... in this case, the Military should allow discovery to clear the air. Its just a BC and some college records. We all know Obama to be honest and intelligent, so there should be nothing to fear about discovery. He has also stated that he wants transparency, so this is a good thing.
Sorry, but it wouldn't hold up. There's an actual paper trail to prove Bush was elected. There is no paper trail showing Obama to be Constitutionally eligible.