Weasels suck eggs. Hence Shakespeare-
The weazel Scot Comes sneaking, and so sucks the princoly egg.
Henry V., i. 2.
“I can suck melancholy out of a song, as a weazel sucks eggs.” —As You Like It, ii. 5. To catch a weasel asleep.
To expect to find a very vigilant person nodding, off his guard; to suppose that one who has his weather-eye open cannot see what is passing before him. The French say, Croir avoir trouvé la pie au nid
(To expect to find the pie on its nest). The vigilant habits of these animals explain the allusions.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894