An idea. Identify H^2(C_1 x C_2, R) with R^k, given the standard Euclidean norm. Now your curves E1, E2, .... are identified with an infinite sequence P1, P2, .... in R^k. You have Ei^2 < 0 and Ej^2 < 0, but (since all your curves are irreducible) Ei Ej >= 0. I imagine this gives you a lower bound on the angle between Pi and Pj for all i,j, which would in turn give you finiteness of the sequence (and indeed an upper bound for the number of such curves, by a sphere-packing bound.) Do you think that works?

An idea. Identify H^2(C_1 x C_2, R) with R^k. Now your curves E1, E2, .... are identified with an infinite sequence P1, P2, .... in R^k. You have Ei^2 < 0 and Ej^2 < 0, but (since all your curves are irreducible) Ei Ej >= 0. Is there such a sequence in H^2(C_1 x C_2, R)?

EDITED to reflect that David Speyer observes that yes, there are infinite sequences of points like this (and that the subspace H^{1,1} of H^2 is what one wants to consider.) David's comment below refers to the version prior to this edit.

Given the existence of such a sequence of cohomology classes, one then asks whether the cohomology classes are represented by irreducible curves, which is what Dmitri wants.