How can Melchisedek be eternal and not be God?

The answer is actually easy.

Let's read the verse that muslims call upon as it being strange or a contradiction.

Hebrews 7:3 He is without father or mother or genealogy, having neither beginning of days nor end of life, but being one with/engulfed/united/connected to the Son of God he continues to be a priest forever.

-This speaks of a divine priest that took human form. His human form was not born of parents for him to "have beginning of day," nor did he die on earth. This is what is meant when saying, he had no beginning and end of days. -In eternity, there are no days and nights, days are only for life on earth.

So was he eternal? Yes, but to understand that, one must know what being “eternal” is.

One can be eternal in two ways.

1) In innate nature. Therefore His existence was always eternal. He cannot lose his eternal nature. Only the God Deity is an eternally existing Nature, no one else. His eternal nature is dependent on no one else, since there is no one above Him.

2) In an acquired nature, given by the Creator. All people have eternal souls in order to live. That soul had a beginning, but it was meant to have no end; this was part of God’s plan; it will either existing eternally in the Presence of God or existing in hell. This given eternal nature, is dependent on God’s power. If God wants, He can cease it at any time; till then, it will exist forever.

   Melchizedek, was a creation made for eternity, which means he was not meant to die, even though God has the power to cut him off from His eternal life source. Therefore, there was a time he did not exist and was created into existence before he came to earth; eternal angels and eternal human souls also didn't exist at one time, but were given existence. 

He was not given birth to like the Son of God was. The Son’s birth was an supernatural one, that we cannot conceive. The Son was always in God the Father, but transitioned in another state within God the Father. This transition within the Father, is called birth and is a mystery.

    Melchizedek is now one with the Son of God and therefore lives eternally with Him, as A PRIEST, because the Son of God in the form of the human Jesus Christ (Heb.5:10, 6:20, etc.) belongs to his order of priesthood, but being above him, as a High Priest!

The order of Melchizedek’s priesthood, is of divine nature and origin and is composed of divine priests, but only Jesus Christ as human with a glorified body is the High Priest among them.

Clarification for muslims: Jesus Christ is God in nature again, as He was before He came to earth. This means He is everywhere, He is omnipresent, but also lives in the glorified human form of Jesus Christ.
-95% of muslims know nothing about the Jesus Christ of Christianity, so all this is all too much for you to take in, but at least you can get the theology right by researching to connect the pieces of the Biblical puzzle that your quran attempts to destroy so that you will be left without the Savior Son of God Jesus Christ.

Here are the major pieces of the puzzle for sincere muslims:

1. http://www.ipertisalithias.gr/index.php?cat=15&id=16&bg=18#article  

2. http://www.ipertisalithias.gr/index.php?cat=15&id=20&bg=18#article  

3. http://www.ipertisalithias.gr/index.php?cat=15&id=10&bg=18#article  

4. http://www.ipertisalithias.gr/index.php?cat=15&id=24&bg=18#article    

5. http://www.ipertisalithias.gr/index.php?cat=15&id=18&bg=18#article

6. http://www.ipertisalithias.gr/index.php?cat=15&id=22&bg=18#article  

7. http://www.ipertisalithias.gr/index.php?cat=15&id=12&bg=18#article  

8. http://www.ipertisalithias.gr/index.php?cat=15&id=26&bg=18#article

   PS. Some muslims also ask, How can Melchisedek be the King of Salem and Jesus be the Prince of peace, is Jesus inferior to him?

---Melchizedek was the King of Salem. While Salem means peace, it was also the place called Zion and therefore Jerusalem:

Psa 76:2 In Salem also is his tabernacle, and his dwelling place in Zion.

So Melchizedek was literally the King of Jerusalem.

Jesus Christ however, is the "Prince" not of a place named Salem, but of peace itself. But why is He called "Prince" of peace and not King? Because His Father, as it should be, is left to be honored as the “King” of peace.