To my knowledge, there is no version of this claim that distinguishes between illegal and legal immigrants. (I'd be happy to be proven wrong.) In fact, it would be silly to claim that illegal immigrants as a group commit fewer crimes per capita than natives, since 100% of them have already violated laws by being on US soil to begin with.

Legal immigrants, on the other hand, are absolutely less likely to engage in criminal behavior than the general population, because they are both selected for positive traits as well as knowing that a criminal record will jeopardize their chances of citizenship.