Hi and welcome to Scibuddies!
Teeth are very nice to have and anything we can do to keep them around longer is a good idea. Tooth enamel is subject to acid erosion from bacteria in the mouth and the sugar in drinks is what feeds the bacteria. But also, as scientists have been recently discovering, loss of tooth enamel can come from over-consumption of acidic drinks such as juices and colas and even some kinds of wine:
https://en.wikipedia.org/wiki/Acid_erosion
I think focusing on acid rather than sugar is the way to go. If you used sugar, you would have to set up an artificial mouth with dental bacteria that produced acid to see if it affected teeth and that would take a long time.
And yes, you will need to measure the pH of your test liquids for comparison of their effects. I'm not sure if an egg is the best thing to test, however. Ideally you would use a substance that would be exactly like tooth enamel. Did you save any of your baby teeth? Can you get some teeth from an animal skull to test? I think that might be more logical than an egg since your hypothesis is that acidic drinks cause erosion of tooth enamel.
I think the best way to measure the effect of an acidic drink on a tooth is to weigh it on a balance that is accurate to at least one milligram (e.g.,
https://www.walmart.com/ip/Smart-Weigh- ... UzEALw_wcB)
Accuracy to a tenth of a milligram is better but such scales are more expensive.
I don't know how long you would need to leave a tooth in an acidic drink (you could use vinegar as one of the test substances because that has a pH lower than fruit juices) in order to see a difference in weight. You could try it for four hours and if you don't see any change in weight then let it sit for another four hours and weigh again. If there's still no change then let it go overnight, 12 more hours or so, and weigh again. Eventually the acid has to eat into it and dissolve some of the enamel.
Try to keep your solutions fairly warm to speed up the process--at least 75F (24C). Also, try to get enough teeth so that you can use three for each test solution. That way you can get an average that is more accurate than a single reading.
I hope this all helps. When you have more questions, we'll be here...
Good luck!
Sybee