The one-word spelling, everyone, is synonymous with everybody (see everybody or everyone); the two-word spelling is far less common and serves only when referring to each of a member of something: Every one of the applicants proved stellar, but we could not hire everyone (a simple test is to swap in everybody; if it works, choose everyone).
Pair the word with singular verbs in both American and British English. However, doing so leads to the great conundrum of what pronoun to use when one is necessary. British English writers seem more comfortable applying plural pronouns; American English writers seem to prefer the grammatical logic of singular pronouns. But the lack of a gender-neutral singular third-person pronoun is likely the greatest source of friction in contemporary English usage (see he or she), and everyone is no exception. Whether you use a singular or plural pronoun, someone will be upset. Nonetheless, it is increasingly common to write phrases such as Everyone knows this, don’t they? Where possible, rephrase to avoid conflicts of grammatical number and exclusionary gender. The above example can be rewritten as We all know this, don’t we?