Of those, only seven have no repeating digits nor a "0".
In the permutations, any particular one of the 4 digits appears as the
one's digit of 4 unique numbers. Since seven are products of odd
primes, there must be at least two odd digits, so 6241 (79^2) is
eliminated. Now, six candidates remain.
Since eight of the permutations are divisible by 2, there must be at
least two even digits present. This eliminates four candidates having
3 odd digits. This leaves two candidates: 1849 (43^2) and 3481
(59^2).
I looked at the 12 even permutation of each for divisibility by 16.
Since both 1984 and 9184 are divisible by 16, this eliminates 1849.
So, the digits are 3481.
Checking: only permutation 3184 is divisible by 16. Also, exactly 4
primes are present in the permutations: 4831, 4813, 8431, and 1483. (OK - I performed this last check by googling "is this prime?")
I didn't bother to check the other four (!) constraints.
soln, p&p
