While a hen may be taken as the type of a foolish woman, a hen with her chickens may symbolize Christ, in allusion to Jesus's words in Matthew ch. 23, ‘O Jerusalem, Jerusalem…how often would I have gathered thy children together, even as a hen gathereth her chickens under her wings.’
as scarce as hen's teeth very rare; as hens do not possess teeth, the implication is that something is rare to the point of non-existence. Originally (mid 19th century) a US colloquialism.
like a hen with one chick absurdly fussy and over-anxious.
See also a whistling woman and a crowing hen.