Wearing a mezuzah necklace might provide for attractive jewelry and be a constant reminder and public declaration of one’s Jewish identity and faith, similar to wearing a Star of David. However, it does not provide special protection to the wearer. In addition, a mezuzah has sanctity akin to a Torah scroll and it should treated with great honor and care. Wearing it would almost certainly involve “disgracing” it (such as when going into a bathroom). It’s possible that a person might wear a mezuzah cover without the mezuzah parchment since the cover has no inherent holiness.
That being said, your friend’s idea that a mezuzah provides protection is not completely wrong. In fact, placing a valid mezuzah on your door-post(s) in fulfillment of the mitzvah earns special protection from God for you and for your children.
Jewish tradition teaches that in the merit of the mezuzah God protects our houses. How does this happen? There is no magic involved, and the explanation is relatively simple. Because the degree of God's involvement in our lives is directly proportional to the degree that we are conscious and aware of Him, the more that we are reminded of God's presence, the more of His presence we will experience. The mezuzah is a vital element in creating and maintaining that awareness — and Divine protection is the result.
This idea of the mezuzah and “home security” is related by the Talmud in a fascinating episode:
Onkelos, the brilliant nephew of the Roman Emperor Titus, converted to Judaism and became a disciple of the Sages. Hearing this, Titus sent a brigade of soldiers to bring him back to Rome. But when Onkelos engaged the soldiers in discussion and showed them the beauty of Torah, they converted to Judaism.
Titus then sent another brigade, instructing them not to speak to Onkelos, but with the same result: They too converted to Judaism.
Finally Titus sent a third brigade and instructed them not even to listen to Onkelos. When they were leading Onkelos away, Onkelos placed his hand on the mezuzah and said, "What is that?"
"You tell us," the soldiers said.
"Normally, a human king sits inside and his servants stand outside and guard him. But, the Holy One Blessed be He, His servants are inside and He guards them from outside."
They too converted. Titus sent no more soldiers.
