I pour the vinegar/water/beet juice solution over the eggs while it is still hot, then cap the jar and leave it out until it is cool to the touch.
By doing this, you create a vacuum in the jar which will prevent, coupled with the anti-bacterial properties of vinegar, bacterial growth.
After it cools, you can toss it in the fridge.
If you didn;t do this, no sweat, just throw it in the fridge now.
I DO think that adding the solution hot helps the eggs to pickle quickly (fluid moves from warm to cold objects - law of thermodynamics and such).
You're making me hungry........I don't have any pickled eggs right now either.