Can an area outside of Israel be conquered during an ongoing Milchemet Mitzva (like Eilat was) according to Halacha?
ב"ה Shalom There are two basic aspects to the Kedusha =holiness of Eretz Yisrael . One is its own Kedusha which stems from the fact that Hashem chose this land for the Jewish people.and promised it to the Avot. This is a Kedusha which is never lost regardless of the exiles from Eretz Yisrael. (כפתור ופרח פרק י) The southern border of Eretz Yisrael in the promise to Avraham Avinu is mentioned as "Nehar mitzraim" (בראשית טו"יח) which some explain as being the eastern tributary of the Nile river תרגום יונתן בראשית ט"ו, תרגום ירושלמי במדבר לד:ה,)), , while others see it as Wadi El Arish ( (כפתור ופרח פרק י"א , רס"ג, שו" הרדב""ז אלף ק"ה. Though, the south eastern border is not specified in Bereshit, it is specified in Shemot (כג"לא) "I will set your borders from the Red sea to the Sea of Plishtim…" (which refers to the Mediterranean). This means that Eilat is included in the biblical promise of Eretz Yisrael to Avraham Avinu. (הרב שלמה יוסף זווין , תחומין ', תשמ"ט.) Furthermore, later in history, Shlomo Hamelech conquers Eilat as mentioned in the Tanach. (במלכים א':ט': כ"ו, דהי"ב ח:י"ז), according to the interpretation of Rav Eliezer Waldenberg, Rav Ben-Tzion Chai Uziel , and Rav Shlomo Goren all of blessed memory. See(ציץ אליעזר ג:כ"ג, משנת המדינה)) Moreover, according to the Rambam (הלכות תרומות פרק א:ב) Eretz Yisrael is defined as lands which were conquered by a Jewish king . Since Eilat was conquered by Shlomo Hamelech, this definition of the Rambam again makes Eilat part of Eretz Yisrael. Accoding to Rav Waldenberg (see above) the fact that Shlomo Hamelech conquered Eilat , elevated Eilat to a level of other places which were conquered by those who came out of Egypt. However, according to Rav Shlomo Zalman Oirbach zt"l , wrote, that on the one hand any place conqured by a Jewish king has the status of the borders of "olei Mitzraim " (מעדני ארץ הל' תרומה פ"א ה"ז, אות ד'.) But on the other hand (מנחת שלמה ח"ב סי' מ"ו ) he wrote that this claim is a bit far reaching and did not want to say that Eilat has the status of the borders of "olei Mitzraim. Nonetheless, in regard to keeping a second day of Yom Tov in Eilat, he recognizes the sovereignty of the State of Israel to make Eilat as part of Eretz Yisrael and to keep only one day Yom tov, since it doesn't make sense that in one country under the same rule some would keep one day Yom Tov and some would keep two. In regard to the conquering of Eilat in our time by Tzahal ( which is so appropriate to speak about just before Yom Ha'atzma'ut, Israel Independence day) Rav Waldenberg zt"l (ציץ אליעזר חלק י: א)says based Rav Ya'akov Emden, known as Ya'avetz ,(מור וקציעה סי' ש"ו ) and Rav Kook zt"l (משפט כהן סי' קמ"ד) writes that the Israel government, the Knesset and the army who function as the representatives of the majority of the Jewish people, have the authority to conquer parts of Eretz Yisrael. Rav Shlomo Yoseph Zevin zt"l, in an article written months before the establishment of the State of Israel, (later published in Techumim Vol. 10) , writes that the mitzva of settling Eretz Yisrael, which was impossible to do throughout the years of exile, will never be more greatly fulfilled until the establishment of the State of Israel. The other aspect of the Kedusha of Eretz Yisrael relates to the mitzvot which are dependent upon Eretz Yisrael, Mitzvot Hatluyot Ba'arets, Trumot and Maserot, Shemitta . In that regard the status of Eilat is different, which requires another discussion, which perhaps will be gotten into at another time. All the Best Happy Yom Atzma'ut .