Matthew 5:13-16 is for the Israelites ONLY?

You are both correct in certain parts. The context has to be considered. The sermon on the mount from Matt. c.5 all through c. 7 was given to a multitude of people, including His disciples before His crucifixion. So, the law and the prophets were still in effect, and was referenced in Matt. 5:17-18 in two ways.

"17 Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.

18 For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled." (KJV)

The "law and the prophets" inclusive of the entire old Mosaic covenant was referred to in the OT prophesies as "heaven and earth". (See my post Heaven and Earth Have Passed Away at my blog ShreddingTheVeil.)

The passing away of the old covenant, that which was waxing old and ready to vanish away in Heb. 8:13, the "heaven and earth" agreement between God and Israel was a future event in that generation. It would pass away, be replaced with the spiritually complete new covenant, new heaven and earth agreement after the complete fulfillment of His coming with His kingdom. (All of the posts at my site present the scriptural evidences of His coming in glory and judgment with His kingdom in A.D. 70.)

There was a time of transition between the cross and the coming of the kingdom, and it mirrored the exodus from Egypt and the 40 year period to the entry to the promised land of Canaan.

Christ was telling the people a great message, clarifying much that the Pharisees and Sadduccees had twisted and misconstrued from the law. All of that sermon was said to people under the conditions of the old law, but looking forward to the new covenant that would apply to all people when His kingdom was fully established.

The trick is that many people do not recognize that His kingdom was established at the destruction of the old sacrificial temple in A.D. 70.

Christ received His kingdom legally at His ascension to the Father in 30-31 A.D. (Rev. 5:6), because He had to go that far country (heaven) to receive His inheritance (John 19:12). But the fullness, or completeness of the establishment of His spiritual kingdom had to wait that 40 years until the gospel of Christ had been preached throughout the known world (the province of Judea and Roman empire) before that spiritual promised land was brought into full existence. (See my post The Promised Land: Between the Cross and the Kingdom again at ShreddingTheVeil.)

So everything that Christ was teaching the Jews, the lost sheep of the house of Israel, was to re-orient their thinking in line with God's original intent in His word, and would equally apply to all of His children under the new covenant of the gospel of Christ.

Since it applied to the Jews at that time, it is also our example today. All who have been baptized into Christ are counted for the seed of Abraham (Gal. 3:26-29), and as such all who are in Christ are now Israel. The definition of Israel changed from the fleshly circumcised to the those of the spiritual circumcision (Rom. 2:29). (See my post God's Definitions - Part I: The House of Israel at my site above.)

Therefore, all of Christ's teachings apply to all of spiritual Israel. They actually apply to all mankind as all mankind are going to be held to His word even if they do not believe Him.

"Ye are the salt of the Earth" is just as important today as when Christ said it: This verse is a warning and it should be one of the most important verses of our era; especially as we watch the near demise of Christianity all around us and churches closed and used by other faiths.

The salt losing it's flavor can be compared to Christians becoming lazy or as the Bible puts it: Becoming barren and unfruitful (2Pet 1:8). If we are the salt of the earth we should be expected to be seasoning the nations of the earth with our Christian Flavor. But this is hardly the case as less Christian are willing to stand for what they believe and to become full-time career missionaries as we let more and more churches close across Europe and sit and watch the United States and Canada follow closely behind.

Nearly every country in the Middle East was a Christian stronghold at one time, but as the centuries passed their faith dwindled and they were overcome by another more zealous faith, and because they didn't have the dander to resist the enemies of the cross, or because it was too much trouble to pray and go to church, or because their women didn't know how to dress properly; then God turned them over to faith that requires them to pray 3 or more times a day, that obligates them to attend the religious service, and that requires women to wear excessive apparel.

Interestingly Spain was among those countries that were overtaken by the yoke of Islam, but the people of Spain rekindled their faith and re-embraced Christ and resisted their invaders. More interestingly history tells us that God chose Spain to be the "salt of the earth" to the New World, and set her sails to the New World to established the Christian Faith every people she conquered - 500 years later the Old Spanish Dominion still adheres to Christian Catholicism, and everywhere missionaries go where the Spanish have trodden (whether in Latin America or the Orient, they find a solid Christian foundation and a people ready to embrace evangelical Christianity.

But not all Old World Christian countries that "lost their flavor" and "were trampled under foot" by attacking armies have as nice as story to tell as that of Spain, none other has fully returned to their Christian roots and are still under an oppressive bondage.

Sadly, Europe the U.S. and Canada are also quickly becoming "flavorless" and in the next 50 to 100 years may soon join the way of Middle East. But it doesn't have to end that way, because again history tell us that less than 100 years ago, when Hitler, Mussolini,and Hirohito threatened to eradicate Christianity from the Earth, it was the Christian nations that rose up resist and defeat the man worshiping armies. I met many an old man who said they lied about their age to be able to go and join the fight - I've seen many graves of boys who never came home to claim the victory they won.

Today Christians are free to travel the Earth to "salt" it with the words of Christ; those who choose to do so, will find that it is already "salted" with little white crosses at battle field grave yards of those who, like John the Baptist, gave all to prepare the way.

You are encountering a theology called Dispensationalism.

Plenty of sites will give you an overview, and many better than Wikipedia, but a key part of their theology is that God dealt with Man in distinct ways in distinct times. Most hold that the change from the 'Old Covenant' to the 'New Covenant' occurred only after the death of Jesus (usually at Acts 2) and that until then all teaching - including the teaching of Jesus - was under the Old Covenant. The Old Covenant was superceded by the New, and its teaching made irrelevant.This irrelevant teaching includes the teaching of Jesus.

I'm not going to get into the rights and wrongs. Dispensationalism is widespread in the United States, and results in significant numbers of people who call themselves Christian yet believe that the teachings of their God in the person of Jesus are irrelevant to them, and indeed were irrelevant only a few years (or even hours) after they were taught. It's easy to work out the implications of that in the current climate.

I'm afraid I'm going to have to leave finding Biblical arguments against this position for yourself, but you might start asking them for the Biblical support of their claims. If you are part of a church that does not hold dispensationalist beliefs, talk to your church leaders about this theology and the church's attitude to it.