In some contexts, there is very little difference between anyone and everyone:
Correct: Everyone benefits from a just government. Correct: Anyone benefits from a just government.
Sometimes they are not interchangeable:
Incorrect: The new CEO knows anyone in the business. Correct : The new CEO knows everyone in the business.
But in this case OP is right, or rather, duolingo's sentence doesn't work in english. "Anyone" is used in questions "Is there anyone here?" or negative sentences "There isn't anyone who will like this." This is not a question, nor a negative so it means the same thing as "everyone."
Again I don't think these sentences are being translated to English well. There must be some difference between "anyone" and "oricine" and it's derivatives because these English sentences do not make sense. It would be great if a Romanian speaker could provide some insight.