This is just my theory, but I think that the basic gene that produces Geostigma lies dormant in the blood of all who carry Jenovah cells. When Kadaj says he sent out "the invitations" it was some kind of biological stimulus( like a pheromone) into the air to trigger Geostigma's activation( which is also, I think how the clones know when to go to a "reunion"). Cloud is a Sephiroth clone. I think all of the Sephiroth clones could get Geostigma, while those who do not have Jenovah cells can't (does this mean that Rufus Shinra is a Sephiroth clone? Do the math;)).
Edit: Since Geostigma is an infection of the planet's lifestream by Jenova cells and, up until the end of the game two years prior, everyone had been using the lifestream as a power source via mako, my guess is that anyone is susceptible to infection. Considering the kids of Edge probably lived in the slums of Midgar prior to the film, it seems likely that they were exposed to mass amounts of mako energy. Anyone else overtly exposed--and Cloud certainly qualifies--probably ran a high risk of infection.