"Is everything alright" is something you might ask a person who looks like they are sad or hurt or otherwise having a bad day. For example "you look tired today. Is everything alright?" It's a very common phrase. "Is anything alright" is a far less common phrase. One might use this if everything seemed to be going badly as a way of saying "so many things are going badly -- is anything alright?" (with an emphasis placed on the "anything").

"Is everything alright?" you ask as a usual question about someone's state when you're dubious about it "Is anything alright?" you ask, f.e. when everything went far too bad and you're asking in other words:"Is At Least anything alright (besides this mess that surrounds you)?"