They use this gene to look at what's going on in mice who are intentionally given the genes for a genetic anomaly homologous with one present in humans. That does not mean that all mice that are given GFP coding regions have one of the controlled genetic problems - they can make healthy GFP mice just as readily as inflicted ones. It's even easier.
The genetic problems you're thinking of are likely located elsewhere on the genome - probably not even the same chromosome - they have nothing to do with GFP.
what picture are you looking at? I'm sure if they know enough to alter genomes, they know what species they're working with.
holy heck, people.
i guess you didnt watch the whole show that included this info directly from the first place to even sequence the rats genes.