Some eyebrows were raised on hearing the answer to this question.
I mentioned this to the question setter, who cited Brewer's Dictionary of Phrase and Fable as his source.
My copy of Brewer's defines Posse as "a body of men, especially constables, who are armed with legal authority".
I had previously understood a posse to be a group of men assembled in a cowboy film to track down and capture someone who had fallen foul of the law. The need for members of the public to be enlisted in this way was a direct consequence of the lack of a standing police force. Brewer's mention of constables would certainly appear to bring the usage closer to police officers, but Brewer defines Constable as "master of the imperial stables [in the Byzantine Empire] ... hence ... an official of a royal household or a military commander ... also the governor of a fortress ... later [from Tudor times] a policeman, as officers appointed to keep the peace".
There seems to be a confusion here between the words "constable" and "policeman" (or "police officer"). As Brewer's makes clear, the two terms are far from synonymous; the use of the word 'constable' to refer to a policeman is almost an afterthought.
Crucially, I don't believe that when Brewer defined a posse as being made up of constables, he was thinking specifically of policemen.
The Oxford [Online] Dictionary defines a posse as "A body of men summoned by a sheriff to enforce the law" (historical US usage), or (in British usage) "The body of men above the age of fifteen in a county (excluding peers, the clergy, or the infirm), whom the sheriff could summon to repress a riot or for other purposes."
The word 'posse' itself comes from the Latin legal phrase posse comitatus, which means 'strength of the county'. Brewer's concurs with the Oxford, explaining that it means "all male members of the county over 15 years of age ..."
The question setter also pointed out that "quite a few collective noun sources" give 'posse' as the collective noun for policemen. I can confirm that this is the case, but in my book (as regular readers of this website will know) that doesn't necessarily mean it's right. Given the above discussion regarding Brewer's and the Oxford Dictionary, it seems to me that whoever first defined 'posse' as the collective noun for policemen was being more than a little creative, and to disallow such words as 'squad' or 'force' is rather unfair.
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