I assume the habit of some hens when kept in a 'free range', or natural system, to lay eggs hidden away under bushes etc, rather than in the nest boxes in the hen house.
The second one is more difficult, in my mind it means that if she doesn't work hard enough, he'll throw her out.
I don't know how old the song is, but in days gone by, women were regarded as chattels, and were the 'property' of their husbands. The married woman's property act wasn't passed in England till 1882, so a song wouldn't need to be very old, for this meaning to be correct.

JM