As far as I can tell, their reasoning is:
The military allows gays.
Therefore all soldiers are gay.
Therefore we should celebrate the death of any American Soldiers, and make asses of ourselves at their funerals.
They truly are nutjobs. Being both rabidly anti-gay, AND rabidly anti-military, they have virtually zero public support in the U.S., and I guess they consider that an honor.
They picketed the funerals of the Boston Bombing victims. They picket football games, concerts, other churches and even Comic-Con.
Even the Ku Klux Klan is disgusted by Westboro Baptist.