The only time I had chain drop issues was when I replaced my chain and mistakenly put it on backwards. I realized that's what I probably done after I had already taken it in to the LBS. They confirmed that was the problem.

I have eTap with Q rings on my road bike with no issues what so ever. The derailleur shouldn't even be hitting the chain I imagine so it's either the ring or the chain. I can only assume you are using new rings and chain. If the rings were older and overly worn it could be an issue as happened to a mate of mine with a new chain older round rings at IM Frankfurt sending him to the deck. Being a brand new chain with too much grease, wax or whatever on it can cause issues too. With a cloth wrapped around the chain, spin the cranks and clean off all the excess lube for a good long period and try again. If the rings are new I would try a different chain first as a process of elimination. If you have a quick link try eliminating that too.

What kind of chain and cassette are you using?

I generally run the Sram 1170 cassettes on training wheels and 1190 on my race wheels but I have never had an issue running a Shimano cassette. I always run Shimano Dura Ace 9000 chains.