Bird eggs cover over half of nest on tower's rear (7)

I believe the answer is:

rooster

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'bird eggs' is the definition.
'rooster' can be an answer for 'bird' (I've seen this before). I'm unsure of the 'eggs' bit.

'cover over half of nest on tower's rear' is the wordplay.
'cover over' becomes 'roof' (I've seen this in another clue).
'half' means to take half the letters.
'of' becomes 'o''.
'nest on tower's rear' becomes 'ster' (I can't explain this - if you can you should give a lot more credence to this answer).
'roof' halved is 'ro'.
'ro'+'o'+'ster'='ROOSTER'

Can you help me to learn more?

(Other definitions for rooster that I've seen before include "Morning caller" , "male with comb" , "Cockerel (mainly US)" , "Adult male domestic fowl" , "Early-morning caller" .)

I've seen this clue in The Daily Express.

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