Being the scientist that I am (not to mention always looking for a short cut!), I decided that, instead of following the instructions, I would conduct an experiment. What would happen, I wondered, if I combined the steps of boiling the dye matter with cooking the eggs? If this worked, it would be an energy-saving technique!
I put eggs, cold water, vinegar, and Red Zinger tea bags into a pot on the stove, brought it to a boil, and let it simmer. I was concerned that boiling the vinegar might create a strong vinegar aroma in the kitchen, but it didn't. After 15 minutes of simmering, I took the eggs and dye off the burner. No change in color at all! Oh, well, I figured, at least I can have egg salad for lunch. I drained the eggs and plunged them into cold water. To my surprise, THAT'S when they changed color - to a mottled light grey-brown.
OK, the first batch came out not very pretty - interesting, but not pretty. Was I deterred then? No! I was fascinated! I decided to try the same experiment with beets. Again I put eggs, cold water, and vinegar into a pot. This time I added the beet juice drained off a can of beets. Brought it to a boil; let it simmer 15 minutes. Results: a very slight change in color to a pale, pale pink.
Conclusion: The shortcut doesn't work! At least not with Red Zinger or beet juice from a can.
Below are the nice egg salad and chilled beets I'm going to have for lunch!