Tree having cuddly extremities to squeeze (7)
I believe the answer is:
cypress
'tree' is the definition.
(cypress tree is a kind of tree)
'cuddly extremities to squeeze' is the wordplay.
'extremities' means to remove the middle letters.
'to squeeze' becomes 'press' ('press' can be a synonym of 'squeeze').
'cuddly' with its centre removed is 'cy'.
'cy'+'press'='CYPRESS'
'having' acts as a link.
(Other definitions for cypress that I've seen before include "Evergreen, mourning symbol" , "Tree with funereal associations" , "Evergreen tree" , "one rooted in the Med?" , "One living in churchyard" .)