I also did some checking around to see if I could find the identity of "J. C.," and I could not. I checked some scholarly editions of Wheatley's work and did not find anything. Usually a scholarly edition would provide this kind of information.
It was common in English-language poetry of the eighteenth century not to identify many people written about in poems. Often initials were used, as in this poem, or sometimes pseudonyms were used (for example, "Eliza," "Alexis," etc.). Such pseudonyms were often classical in flavor and reflected the influence of Roman poets, who were deeply admired. Wheatley's reference to "J. C.," then, would not have seemed unusual.
Wheatley had a volume of poetry published in England during her 1773 visit in the company of Nathaniel Wheatley. Judging from the date of publication, her detached tone as she muses over the questions of "the parent," and the later date of her own marriage and family, it is probable "J. C. an Infant" was someone she met in England with whom she had only passing acquaintance and who did not want names used.
I, like the previous poster, looked for an answer. The only thing that I can come up with, outside of that of trying to find out who "James" is given the poem references that name, is Jesus Christ.
Wheatly's poetry, "Poems on Various Subjects, Religious and Moral", speaks to the fact that she was highly religious. Given that she was also a slave, one could infer that her reference COULD be made based upon the fact that some slave owners did not allow talk of religion by their slaves. She could have been masking her religious theology.
Another idea is that she is simply mourning the death of a child, but raising it up at the same time given "cheerful resign at divine command."
This is a reach!! I honestly found no reference to a JC in the life of Wheatly other than James Madison and James Bowdoin, neither who have the initials JC.
I have searched and can find the poem on many web sites, but I have been unable to find any commentary on it. The poem was published in a collection of Phyllis Wheatley's poetry entitled Poems on Various Subjects, Religious and Moral, and not even that book contains commentary or marginal notes. The book does, however, include several similar poems, all written in memory of a person who had died. There is a clue in the poem that the infant's first name was James:
"Where lies my James? tis thus I seem to hear
the parent ask"
Other than his first name, we'll probably never know who the infant James was.