No. That would obviously be ridiculous, chickens would not be an effective method of detonating a nuclear weapon. That said... there was a proposal that they may have been used as an element of the Blue Peacock nuclear landmine or Atomic Demolition Munition (ADM).
With fears of Soviet forces sweeping through Germany ADMs would have been used to destroy critical installations and infrastructure to prevent them falling into enemy hands, the resulting contamination being an added 'benefit' in such a scenario. First proposed in the mid-1950s Blue Peacock would have been a 10 kiloton weapon based on the Blue Danube free-fall nuclear bomb (the Ministry of Supply at the time used Rainbow Codes consisting of a colour and a noun) housed in a pressurised case, detonated either on command by wire (a very long wire!) or by a clockwork timer.
The project was shelved in the late 1950s, not least due to the political implications of irradiating large swathes of allied territory, but an inert prototype survived to become part of the Historical Collection of the Atomic Weapons Establishment (AWE), described in an article by David Hawkings in the Summer 2002 edition of Discovery, the AWE house magazine.
Where do the chickens come in? Well, the warhead had to be kept within a specific temperature range and according to Hawkings "... environmental trials suggested it might not have survived the rigours of a mid-European winter. Keeping Blue Peacock warm involved swathing it in glass fibre pillows." In 2004 the National Archives opened a new exhibit, Secret State, involving newly declassified material, including documents on Blue Peacock that forecast an armed life of 4 days with 4" of insulation in temperatures of -25°F. This could be extended by:
"(a) Increasing the thickness of the insulation beyond 4". 6" would be a practicable maximum but would be clumsy.
(b) Incorporating some form of heating independent of power supplies under the weapon hull in the emplacement. Chickens, with a heat output of the order of 1,000 B.T.U. per bird per day are a possibility."
By a quirk of timing the exhibition opened on April 2nd 2004, so previews appeared on... April Fool's Day. This led to Patrick Barkham's article in The Times of Thursday April 1st "Is today the day to reveal the chicken powered nuke?", outlining Blue Peacock and the warming chickens, including a quote from Professor Peter Hennessy, curator of the Secret State exhibition: “It is not an April Fool. These documents have come straight from the archives at Aldermaston. Why and how would we forge them?”, and one from Tom O’Leary, head of education and interpretation at the National Archives: “It does seem like an April Fool but it most certainly is not. The Civil Service does not do jokes. The reputation of the National Archives would be damaged if we made this up.”
The chicken element naturally features in various Weird Weapons or Strange History books and articles, headlines not always quite capturing their role (they didn't exactly power it, as per The Times), it seems more of a vague proposal than a serious proposition, but the bomb itself certainly existed, at least to prototype stage.