Does the prohibition of leaving Eretz Yisrael apply to visiting portions of the Central Negev? I am specifically referring to Mitzpeh Ramon/Machtesh Ramon. I am unclear whether this area is included in the area of the boundaries of Eretz Yisrael listed in Parshat Masei. Additionally, in general, if an area is outside the Biblical boundaries of Eretz Yisrael (as listed in Parshat Masei), but was part of the Kibush of Olei Mitzraim (e.g. in Shlomo Hamelech’s days Bnei Yisrael inhabited Etzyon Gever (i.e., Eilat)), would it be permissible to leave Eretz Yisrael to visit those areas?

Tosfot in Gittin 2:1 says that the prohibition of leaving Israel is from the boundaries of Kibush Olei Bavel. It says that southern of Ashkelon isn't in the Kibush of Olei Bavel and visiting there is not permitted, including, of course, the Central Negev and Mitzpeh Ramon. But many Poskim agree that after it was conquered the prohibition to visit those places became void. One may ask If the places you mentioned are part of Kibush Olei Mitzraim, and that depeneds on the famous Machloket among the Rishonim if Nachal Mitzraim mentioned in the Torah is The Nile, as written in Rashi and other Rishoinm, or other that claim it's referring to Wadi El-Aris. According to the first opinion the entire Sinai desert is part of Kibush Olei Mitzraim, but according to the latter the area is much smaller, and the places you've mentioned are not included even in Parashat Masei. Even so, not all is clear since the Achronim mentioned that the Rosh who says Nachal Mitzraim is Wadi El-Aris says on the other hand, that Maale Akrabim is Akaba at the Red Sea accroding to some researches. The Kaftor Vaferach (chapter 10) writes that today there is no difference between Kibush Olei Mitzraim and Kibush Olei Bavel and there is no Minhag about the Kedusha. Regarding the last question I believe Rabbi Yisraeli wrote that all conquered by Olei Mitzraim is in the borders of Parashat Masei. Rabbi Chaim Katz