Is "cannot not say" standard English?

Solution 1:

"cannot not say" would only rarely be used in English, and only in very specific circumstances. In particular, this is not a simple double negative. "cannot not" does not mean the same as "can", it means "must". It also carries a connotation that not saying is the expected or default action.

"I must say precisely what attracted them to the part of Giuliette" is probably not the message you are intending to convey here, so I expect that there is one extra negative in your sentence.

• "I can take action" means "I am able to take action"
• "I cannot take action" means "I am not able to take action"
• "I cannot not take action" means "There is some reason that not taking action is expected here, but I must take action"

Solution 2:

"I cannot not say X" means "I am unable not to say X" or, in other words, "I am not able to refrain from saying X".

When I heard that ruffian yelling obscene curses at a small child, I just couldn't not say anything, you know?

"I cannot not say precisely what attracted them to the part of Giuliette" is grammatical, and it states "I am unable to refrain from saying precisely what attracted them to the part of Giuliette".

If that is not the intended meaning, then it is an inappropriate selection of grammar.