Here is an article I took from Mind Our English
WHICH is the better sentence? We should never ever give up on everything that we do. OR We should never ever give up on anything that we do. I think it is the second sentence, but why is that? – MOE reader
You are right. The second sentence is better. This is because “anything” is usually used in a negative sentence, and both your sentences are negative.
“Everything” is better used in a positive sentence, e.g. “We should be wholehearted in everything we do.”