All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His Slave and Messenger.
A non-Muslim should not be hired to clean a mosque if there are Muslims who can do that job because of the difference in opinion among the scholars about the ruling of a non-Muslim entering the mosque; some scholars are of the view that it is permissible while others deem it impermissible.
If there is no Muslim who can clean the mosque and it is guaranteed that the non-Muslim will not defile the mosque, then there is nothing wrong in him entering the mosque and cleaning it. However it should be ensured that he will not touch the Mus'haf (hard copy of the Quran).
What we believe to be the preponderant view, is that it is permissible for a non-Muslim to enter the mosques in case of need. This view is also the opinion adopted by the Standing Committee. They said about the ruling of a non-Muslim entering the mosque: "The correct view is that it is permissible for a non-Muslim to enter the mosque if there is a benefit or a need to do so [..] The Prophet tied Thamaamah ibn Athaal Al-Hanafi in the mosque before he became a Muslim and he, sallallaahu 'alayhi wa sallam, also received a delegation from Thaqeef and a Christian delegation from Najraan in the mosque before they became Muslims. This was because of the many benefits resulting (from them entering the mosque), such us hearing the sermons of the Prophet and seeing those who were praying and reciting the Quran, and other great benefits that only those who stay in the mosque can obtain." [End of quote]
Allah Knows best.