Is there any Hadith in which the Prophet instructed his sick companion to drink the urine of a camel? If it is as mentioned, what is the Hadith and why did the Prophet give such an instruction (Peace be upon him)?
We live in the West and this question has been posed to us by non-believers.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

Al-Bukhari reported from Abu Qulaba from Anas that he said,

"The climate of Madeenah did not suit some people who came from 'Ukul or 'Uraynah. So, the Prophet ordered them to follow his shepherd, i.e. his camels, and drink their milk and urine (as a medicine). Thus, they followed the shepherd, (i.e., the camels) and drank their milk and urine until their bodies became healthy. Then they killed the shepherd and took the camels away. When the news reached the Prophet he sent some people in their pursuit. When they were brought, he cut their hands and feet and their eyes were branded with heated iron bars and they were left in Al-Harra until they died in that state. "

Abu Qulaba said, "They reverted to disbelief after conversion to Islam, stole and killed. In fact, they fought Allaah and His Messenger."

There is evidence from this Hadeeth that the milk and urine of camels is a treatment for some diseases. This is mentioned expressly in the Hadeeth narrated by Ibn 'Abbaas from the Messenger of Allaah The Almighty that he said: "The milk and urine of camels is a cure for those who suffer from (stomach) indigestion." [Ahmad]

An-Nasaa'i, may Allaah  have mercy upon him, reported from Anas about the people of 'Ukal and 'Uraynah, "When their skins became yellowish and their bellies got very big, the Prophet sent them to his camels and ordered them to drink its milk and urine."

Ibn Al-Qayyim considered this an 'authentic Prophetic medicine'.

Allaah Knows best.