Question: Is one obligated to shave before Yom Ha-Atzmaut (since there is a custom not to shave during the period of Sefirat Ha-Omer)?
Answer: Yes, just as one shaves all year long to honor Shabbat, it is a mitzvah to shave in honor of Yom Ha-Atzmaut. Obviously, if a person does not shave his beard at all, he does not do so, but if he generally does so for Shabbat, then he also shaves for Yom Ha-Atzmaut. The same applies to getting a haircut. Ha-Rav Yitzchak Nissim, former Chief Sefardic Rabbi of Israel, rules in Shut Yayin Ha-Tov (vol.2, Orach Chaim #11) that it is it is permissible to get a haircut and shave in order to honor the day based on various authorities who allow haircuts and shaving during Sefirat Ha-Omer for similar reasons. For example, Ha-Rav Chaim Palagi (Sefer Moed Le-kol Chai siman 6, pg. 36) wrote that in his city of Izmir there were families who experienced miracles during Sefirat Ha-Omer and established them as days of joy and thanksgiving like Purim and cut their hair and shaved on them. Ha-Rav Nissim therefore concludes that on a day when a miracle occurred during Sefirat Ha-Omer - like Yom Ha-Atzamaut - when the State of Israel was declared after 2000 years it is certainly permissible, and even a Mitzvah, to have a haircut and shave on that day (see also Ha-Rav Yaakov Ariel, Chief Rabbi of Ramat Gan, in Siddur Ga’al Yisrael for Yom Ha-Atzmaut p. 32, #13. And Ha-Rav Shlomo Goren ruled that it was permitted for soldiers to have haircuts and shave on Yom Ha-Atzmaut as indicated in a Command of the General Staff of the Chief Rabbinate of Tzahal #34.0207 brought in Hilkhot Tzava of Ha-Rav Zechariah Ben Shlomo, p. 861). The Chief Rabbi of Yerushalayim, Ha-Rav Tzvi Pesach Frank, also gave permission to trim one's beard on this day (See Luach "Shana be-Shana" published by Heichal Shlomo, 5752, p. 145. For those who disagree, see Ha-Rav Ovadiah Hadaya in Shut Yaskil Avdi 6:10 #2 and Ha-Rav Yosef Dov Soloveitchik in Nefesh Ha-Rav, p. 94). I remember that some students once asked our Rabbi, Ha-Rav Tzvi Yehudah Ha-Cohain Kook, before Yom Ha-Atzmaut if it is permissible to shave for this day. Our Rabbi did not answer. Later he responded: "Tomorrow there will be an exam on your faces." On Yom Ha-Atzmaut he saw that some students had been strict with themselves and had not shaved. He said of them: "Their faces show their character" (Yeshayahu 3:9, i.e. they had not absorbed the true joy of the day). He added: When there is faith, there is joy, and when there is joy there are no halachic doubts. How long will we speak out of both sides of our mouths (see Melachim 1 18:21)?! Do we believe in the Revealed Redemption or not?!"
Q: When should one get a haircut or shave, since Erev Yom Ha-Atzmaut is Remembrance Day for the Fallen Soldiers of Tzahal?
A: One should do so a little bit before the holiday begins. For example, if Ma'ariv is at 7:00 PM, he should shave at 6:00 PM.