From the American Heritage Book of English Usage:
anyone is always singular. Anyone and anybody are singular terms and always take a singular verb.
anyone or any one. The one-word form anyone means “any person.” The two-word form any one means “whatever one (person or thing) of a group.” Thus, Anyone may join means that admission is open to everybody. Any one may join means that admission is open to one person only. When followed by of, only any one can be used: Any one (not anyone) of the boys could carry it by himself.
anyone instead of everyone. Anyone is often used in place of everyone in sentences like She is the most thrifty person of anyone I know. While this construction has been used for centuries, 64 percent of the Usage Panel in an earlier survey found this sentence unacceptable in writing.