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.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894