Can genetically modified genes jump to bacteria in the eater's intestine?
No.

There is nothing special about a piece of GM DNA when compared to any other random piece of DNA. If this phenomenon happened at any detectable level, we'd have found eukaryotic DNA in bacterial genomes/plasmids long before the introduction of genetically modified crops. And that would be front page news in the field of horizontal gene transfer! Which has many talented people in it!