Secure multiparty computation of number of 1s times number of 0s in a shared string
This answer is about feasible solutions based on homomorphic encryption that is NOT fully-homomorphic, as the latter may be extremely inefficient (if there are efficient fully homomorphic cryptosystems that are comparable with those provided below in terms of efficiency, I'd be glad to hear about them).

Since you only need one multiplication then there are solutions that are potentially less expensive than fully homomorphic encryption: [1] and [2]. The latter works on encrypted bit-decompositions of the input so it will need a bit-decomposition protocol like [3] and [6], but the former works on whole values. Just for completeness, the former has been extended to $d$-operand multiplication in [4], even though the OP may not need this. These solutions are non-interactive and should work in the two-party case. 

If you have more than two parties and could afford some interaction then [5] provides a "secure multiplication gate" which is potentially more efficient and allows unbounded number of multiplications. It works basically by converting the homomorphically-encrypted values to some sort of secret-sharing, multiplies the result (interactively), then convert it back to homomorphic encryption.

[1] Evaluating 2-DNF Formulas on Ciphertexts

[2] Non-interactive cryptocomputing for NC1

[3] Unconditionally Secure Constant-Rounds Multi-party Computation for Equality, Comparison, Bits and Exponentiation

[4] Additively Homomorphic Encryption with d-Operand Multiplications

[5] Multiparty Computation from Threshold Homomorphic Encryption

[6] Efficient Binary Conversion for Paillier Encrypted Values