The ability to intercede on behalf of others is a gift given by God, and so He can give it to anyone He pleases. Here is how the Catechism of the Catholic Church describes prayer of intercession:
Since Abraham, intercession—asking on behalf of another—has been characteristic of a heart attuned to God’s mercy. In the age of the Church, Christian intercession participates in Christ's, as an expression of the communion of saints. In intercession, he who prays looks “not only to his own interests, but also to the interests of others,” even to the point of praying for those who do him harm (no. 2635).
All that is required is that God inspire the intercessor to make the prayer. Hence, it is not at all limited to Catholics or even to Christians.
As far as the deceased are concerned, the ability to intercede on behalf of those on earth is not limited to those who have been officially canonized. Anyone who is in friendship with God has this ability, even those who were not in full union with the Church. Hence, if a Protestant or non-Christian died in friendship with God, and hence was saved, then he can intercede, just as any other saved soul can.
(As the parable of the Rich Man and Lazarus suggests, presumably, even the condemned could, in theory, intercede for us, if God gave them that power, but St. Augustine suggests that it would be for selfish motives [see no. 17]; e.g., the rich man interceded because the presence of the his brothers would only increase his torment.)
Note that canonization and beatification are done chiefly because the person has lived a life of holiness and heroic virtue; that he intercedes for someone and provides a miracle is only a sign that he is already in Heaven.