Can you bust a vessel or vein in your eye by blowing your nose too hard?

Indeed: You have a subconjunctival hemorrhage due to the increased pressure you created in your eye by blowing your nose. Be careful. The redness will resolve slowly but is usually not a major problem. It may be good to check your blood pressure to make sure your pressure is normal. If the redness does not resolve or is quite pronounced, see a physician. Good luck!

Yes: Yes. A sudden strong pressure can rupture a capillary on the surface of the eye. The blood can collect under the clear membrane over the white part and look quite frightening. It is generally not harmful and resolves on its own slowly. This is called a subconjunctival hemorrhage.