This is too late, as the 2016 Diversity Visa registration ended Nov. 3, 2014, but...
In the 2016 Diversity Visa instructions, it states that
you may choose your country of eligibility as [...] the country of birth of either of your parents if you were born in a country in which neither parent was born and in which the parents were not resident at the time of your birth.
So, in your case, you can only choose Egypt as your country of eligibility if neither of your parents was a residence of the UAE when you were born.
If that's the case, and you really do have a choice, the documents on the State Department's Diversity Visa Program Statistics page have enough information to compare the relative odds in past lotteries of being chosen in those countries. (Don't forget, things might be different in future years.)
For example, in the 2015 lottery, there were 7,649 applicants from the United Arab Emirates. Including derivatives (spouses and children), the total was 15,696. 195 of these were registered. My guess, from the wording on the web page and from the numbers given, is that this number includes derivatives. (They don't seem to say explicitly.) That would mean 1.2% were chosen.
For Egypt, there were 309,143 applicants, for a total of 763,701 counting derivatives. 4,988 chosen means a 0.65% success rate, lower than that for the UAE.
In fact, no country in 2015 had more than 5,000 selected entrants. Looking at some other countries' numbers, it appears that most of the countries in the Africa region (like Egypt) with well under 5,000 selected have similar success rates to each other, and the same for the Asia region (which includes the UAE). It's the ones near 5,000 that appear to have reduced rates of success.
This shows that the chance of getting chosen in the 2015 lottery was lower for applicants from Egypt than from those in the UAE. Applicants from Egypt were effectively penalized, by a lower chance of success, because of the high number of total applicants + derivatives from that country. Clearly the applicants from the UAE had no such problem.
The numbers from 2014 tell a similar story, with a better chance of success from the UAE than from Egypt.
So in your case, no reason to complicate things. If you feel the most logical country to put on your application is the United Arab Emirates, then just put it on your application. Past lottery results are in your favor.