The issue of Eilat being part of Eretz Yisrael or not, from a halachic perspective, is an old one and has been discussed many times before.

The relevance is really for kohanim, for whom leaving Eretz Yisrael could involve a prohibition of impurity, and for people spending a holiday in Eilat who needs to know how many days of holiday to observe when in Eilat.

The issue is complicated, and truthfully for most people it is practically irrelevant - especially because most people travel abroad - anybody who travels abroad and is not concerned about the impurity of stepping on the ground in New York, London, Paris or anywhere else - shouldn't have a problem in Eilat. The question of 1 or days of holiday observance is relevant.

I would like to see any rav who paskens that Eilat is not part of Eretz Yisrael add a comment to his psak. Instead of just saying its not eretz yisrael, he should add that anybody who travels abroad to other cities should have no problem traveling to Eilat.

Rav Dov Lior recently said that he recommends kohanim not travel to Eilat, because of the question and uncertainty as to its status. The Tzitz Eliezer holds Eilat was conquered by Shlomo Hamelech and is part of Eretz Yisrael, but his opinion is not part of the majority opinion on the matter. Rav Lior adds that the Rambam allows one to travel outside of Eretz Yisrael for the purpose of commerce, and being that Eilat is a port city, it is allowed to travel to it - as the entire city is considered part of the commerce of the port. On the other hand, there is no positive commandment of settling in Eilat, as while one could travel there, it still is not part of Eretz Yisrael. So, Israelites can travel there with no concern, but kohanim should be stringent unless it is necessary.

I always wondered what Rav Moshe Hadaya, the Chief Rabbi of Eilat, would have to say about the discussion but never heard him comment on it (though I am sure he has had a halachic opinion all along)... it seems that Rav Lior's recent comments have caused Rav Hadaya to speak up...

Rav Hadaya said that his uncle, Rav Ovadia Hadaya, a member from the days of the foundation fo the State of the religious council of the Rabbanut and a member of the Beit Din Hagadol, Rav Tzvi Pesach Frank, the Chief Rabbis of Israel - Rav Herzog and Rav Uziel, all have clearly said that once Eilat was captured by the IDF it became part of Eretz Yisrael in every regard. He would add that if we were to announce that Eilat is not a part of Eretz Yisrael, the Egyptians would demand that we should return Eilat to them.

Rav Hadaya also points to Rav Ovadia Yosef for this discussion. He says he does not understand those who make statements just to stand out. Rav Ovadia Yosef traveled to Eilat 30 years ago and Rav Ovadia asked him - asked Rav Hadaya - whether he observes one or two days of holiday. Rav Hadaya responded that he observes one day just like everywhere in Israel. Rav Ovadia's response, according to the testimony of Rav Hadaya, was to turn to those around and say "did you hear what the rav said? I will not talk about this issue."

Rav Hadaya also compares it to some other cities that technically should have the same status, as per the gemara. Lod and Bnei Braq should really be considered chutz laaretz, but nobody would dare suggest that Lod or Bnei Braq be treated like chutz laaretz - so what do they want from Eilat?
source: Srugim and full interview in Erev Erev

I feel vindicated, in a sense, because when recently learnign a connected sugya in the gemara, the topic of discussion naturally turned to this issue, and my opinion expressed (non-definitively) then matches very closely to what Rav Hadaya just said...