More musing than answer. (Technically, it is wrong, because I am adding the coordinates wrong. I think I should be summing n_i -n_j instead of n_i. I will edit in a correction later.)
I imagine that principal means "having the diagonal come from the diagonal of the large matrix". Then a square submatrix is determined by (the entries of the large submatrix, of course, and) a set of n integers, which are the coordinates of the columns and of the rows. Suppose we start the coordinates at zero and the rest are positive integers n_i in ascending order. Then the values of the ith jth entry of the sub matrix (for i less than j) are on the kth diagonal of the large matrix, where k is n_i + n_j. So this question is really a sumset question, as the large matrix is circulant, the entries depend only on the sum (of the coordinates from the n-set) of values.
Gerhard "A Musing Is Sometimes Amusing" Paseman, 2017.12.02.