Yes, the definition of person given by Boethius (De Duab. Nat.) is sufficient (that is, "a person is an individual substance of a rational nature") because substance is individualized by itself (whereas the accidents are individualized by the subject, which is the substance) and so it is reasonable that the individuals of the genus substance should have a special name of their own (for they are called "hypostases," or first substances); and, in a more special and perfect way, the particular and the individual are found in the rational substances which have dominion over their own actions (and hence the individuals of the rational nature have a special name even among other substances: "person").
Thus the term "individual substance" is placed in the definition of person, as signifying the singular in the genus of substance; and the term "rational nature" is added, as signifying the singular in rational substances.
According to the Philosopher (Metaph. v, 5), the word "nature" was first used to signify the generation of living things, which is called nativity. And because this kind of generation comes from an intrinsic principle, this term is extended to signify the intrinsic principle of any kind of movement. In this sense he defines "nature" (Phys. ii, 3). And since this kind of principle is either formal or material, both matter and form are commonly called nature. And as the essence of anything is completed by the form; so the essence of anything, signified by the definition, is commonly called nature. And here nature is taken in that sense.
Hence Boethius says (De Duab. Nat.) that, "nature is the specific difference giving its form to each thing," for the specific difference completes the definition, and is derived from the special form of a thing. So in the definition of "person," which means the singular in a determined "genus," it is more correct to use the term "nature" than "essence," because the latter is taken from being, which is most common.