Hate, like, love, prefer
Hate, like, love and prefer can be followed either by -ing or a to-infinitive. The difference in meaning is often small. The -ing form emphasises the verb itself. The to-infinitive puts the emphasis more on the preference for, or the results of, the action.
How would you like
at you while you're working "FILE THOSE PAPERS FASTER - RUN FROM THE PARKING LOT INTO THE OFFICE - YOU REALLY NEED ANOTHER BREAK!
I found the above sentence with "would you like" (instead "wouldn´t you like") and "someone" instead of "anyone", but the same principle applies here.
That is, the Cambridge rules applies to both examples of "like":
"She wouldn't like anybody to know." would be with emphasis on the preference for the result being that of nobody knowing it
. Or the preference of the action of somebody else not being that of letting others know/anybody know
Whereas, my example (the one I provided) doesn´t focus on the result, or preference that "I/she/he, ..." for some given result, the emphasis is on the verb, the action itself. On something that happens (or it could happen in general). In this case, it takes a gerund form.