I have spoken to people who bring the idea that Bilaam was a prophet and therefore non-Jews can be prophets. But that was specific to the context and hence is no proof. G-d wanted to give the non-Jews a leader of equal spiritual calibre so that they would not have the excuse of not having the likes of Moshe. Does it say anywhere that non-Jews can be prophets? Or anyone can be prophets for that matter?
In his Epistle to Yemen, Rambam writes very clearly that non-Jews can be prophets:
Our disbelief in the prophecies of Omar and Zeid is not due to the fact that they are non-Jews, as the unlettered folk imagine, and in consequence of it are compelled to establish their stand from the biblical phrase from among your own people. For Job, Zophar, Bildd, Eliphaz, and Elihu are all considered prophets by us although they are not Israelites.
Note that this does not mean that anyone can be a prophet though. In Guide for the Perplexed 2:32 Rambam describes the limitations on who can become a prophet:
Similarly there are three different opinions on Prophecy. I will not notice the view of the Atheist; he does not believe in the Existence of God, much less in Prophecy; but I will content myself with discussing the various opinions [on Prophecy] held by those who believe in God.
Among those who believe in Prophecy, and even among our coreligionists, there are some ignorant people who think as follows: God selects any person He pleases, inspires him with the spirit of Prophecy, and entrusts him with a mission. It makes no difference whether that person be wise or stupid, old or young; provided he be, to some extent, morally good. For these people have not yet gone so far as to maintain that God might also inspire a wicked person with His spirit. They admit that this is impossible, unless God has previously caused him to improve his ways.
The philosophers hold that prophecy is a certain faculty of man in a state of perfection, which can only be obtained by study. Although the faculty is common to the whole race, yet it is not fully developed in each individual, either on account of the individual's defective constitution, or on account of some other external cause. This is the case with every faculty common to a class. It is only brought to a state of perfection in some individuals, and not in all; but it is impossible that it should not be perfect in some individual of the class; and if the perfection is of such a nature that it can only be produced by an agent, such an agent must exist. Accordingly, it is impossible that an ignorant person should be a prophet; or that a person being no prophet in the evening, should, unexpectedly on the following morning, find himself a prophet, as if prophecy were a thing that could be found unintentionally. But if a person, perfect in his intellectual and moral faculties, and also perfect, as far as possible, in his imaginative faculty, prepares himself in the manner which will be described, he must become a prophet; for prophecy is a natural faculty of man. It is impossible that a man who has the capacity for prophecy should prepare himself for it without attaining it, just as it is impossible that a person with a healthy constitution should be fed well, and yet not properly assimilate his food; and the like.
The third view is that which is taught in Scripture, and which forms one of the principles of our religion. It coincides with the opinion of the philosophers in all points except one. For we believe that, even if one has the capacity for prophecy, and has duly prepared himself, it may yet happen that he does not actually prophesy. It is in that case the will of God [that withholds from him the use of the faculty]. According to my opinion, this fact is as exceptional as any other miracle, and acts in the same way. For the laws of Nature demand that every one should be a prophet, who has a proper physical constitution, and has been duly prepared as regards education and training. If such a person is not a prophet, he is in the same position as a person who, like Jeroboam (1 Kings xiii.), is deprived of the use of his hand, or of his eyes, as was the case with the army of Syria, in the history of Elisha (2 Kings vi. 18). As for the principle which I laid down, that preparation and perfection of moral and rational faculties are the sine quâ non, our Sages say exactly the same: "The spirit of prophecy only rests upon persons who are wise, strong, and rich." We have explained these words in our Commentary on the Mishnah, and in our large work.
(Friedlander translation, my emphasis)