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.