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?