[momincombatboots]

There are several ways to DNA gets transferred. A piece of DNA enclosed in a bacteriophage could be resistant to acid and self enzymes, also by several mechanisms.

[allmendream]

Not any likely way. The DNA of the things you eat does not get inside your own cells to “muck up” your own DNA.

It just doesn't happen.

Of several legitimate concerns with gene modified organisms, that has to be lower on the list than little green men from Mars.