According to many religious Zionist poskim, there is a special dispensation to allow music on 5 Iyar/Yom HaAtzma'ut. Is there a discussion at all in contemporary religious Zionist rabbinic writings regarding shaving and/or getting haircuts on 5 Iyar (whether officially Yom HaAtzmaut or when the official commemoration was earlier])? Is there a potential distinction made when 5 Iyar falls out on Friday (as this year ; a la the leniencies regarding haircutting/shaving on Fridays more generally)?
R' Eliezer Melamed discusses this in the Zmanim volume of פניני הלכה.
He writes that the generally accepted approach to all of the minhagim of avielut of the omer is to be lenient with those things which would actually reduce the simcha of the day. For example, forbidding music and dancing would reduce the simcha so those are muttar but not allowing weddings wouldn't really make any difference so weddings remains assur.
Accordingly, he says that those who would usually shave before a chag or such like ought to shave before Yom Ha'atzmaut while only someone who looks dishevelled should have a haircut beforehand. On the day itself however, even someone who looks fine can have a haircut.
You can find the entire series of פניני הלכה at http://ph.yhb.org.il. The place where he discusses this particular issue can be found at http://ph.yhb.org.il/05-04-11/. You can find the various opinions he is basing himself off in the footnoes there.
(Note: He makes no distinctions between whether Yom Ha'atzmaut falls on 5th Iyar or not. Earlier in the book, he discusses the conditions and details under which it is nidcheh (most years) and compares it to when Purim is nidcheh. As far as I can tell, everything he writes is equally applicable to when it falls on 5th Iyar or otherwise.)