Why does ant queen keep the eggs out of her hive?
According to various websites and literature I found, 

Starting a new colony

During the mating season, Queen Ant grows wings and mate with male (flying) ant. In most of the species the queen will fly away to a new place where, she will loose her wings and dig a tunnel. She may lay eggs inside or outside because of the humidity and temperature, mostly they wait till spring, so that the humidity is high as well as warm temperature.

For laying eggs she will dig a tunnel inside and close the gates. She won't eat or move and completely avoid light, if the eggs are unhatched or some other problem occurs she will eats her own eggs. After around 8-10 weeks eggs will pop and you will get new worker ant babies. They will form the hive and feed the queen ant.

    

More info: http://antnest.co.uk/colony.html