Are there first-order sentences that are preserved under binary products but not all products?

Question

Are there first-order sentences that are preserved under binary products of structures, but not under all products of structures? I would also like an example of such a sentence.

Answer 1

No, there are no such sentences. Quoting Corollary 6.7.2 on p. 84 of S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57–103 (pdf):

If the class of all models of a set of first order sentences is closed under the operation of taking the direct product of two systems, then it is closed under the operation of taking arbitrary, infinite, direct products.

I assume that by "all products" you meant direct products of any nonzero (finite or infinite) number of factors; the answer would be different if you meant to include reduced products or the empty product $\prod_{i\in\varnothing}\mathfrak A_i$.