‘Any one’ means one person or one thing from a group of people or things. It is written as two words: any one.
A: I want to buy all of these dresses! Which one should I choose?
B: Honestly, any one of those dresses would look great on you. It’s up to you!
A: You’re no help at all!
‘Anyone’ means any person. It’s always written as one word: anyone.
-Did anyone see where I put my cell phone?
-This cafe sells the best bagels in all of Brooklyn. Ask anyone!