I almost ate the whole egg before noticing this gross thing I have never seen before. I feel stomach upset in near future!?
These are a normal part of eggs, albeit an unusually large one. If you have ever broken open an egg and noticed reddish flecks in the albumen or around the yolk, then you have seen smaller versions of this one.
Basically these small red bits are bits of the oviduct of the chicken that layed the egg, and are perfectly edible. There should be no reason to cause gastrointestinal problems from consuming these, other than the not-so-nice-to-look-at component.