A comical blog mentioned “patriotic worship”. It is also hinted at in this article and in this article where it is made clear that patriotic songs such as the ones listed here are sung and they are considered suitable for worship by some groups. It is apparent that within the USA patriotism and Christianity are sometimes conflated. As you can imagine, it is quite difficult to search for information regarding worship and patriotism, especially in connection with the United States. I would like to know how prevalent the singing of patriotic songs is, which denominations are the most likely to use them and does their use peak around patriotic festivals such as Independence Day and Memorial Day?
In the LDS church in the US it is common to sing patriotic songs around Independence Day and sometimes Memorial Day and Veteran's Day too (though I think that's more rare). It is very unusual to hear them sung at church any other time.
Official guidance on the matter:
A few patriotic songs have been included in the hymnbook; with priesthood approval, local national anthems may be added. Members may stand for national anthems in Church meetings according to local custom and priesthood direction.
Mormons have a unique feeling of US patriotism because they believe God founded the country to provide a free place for him to restore his church (the LDS church).
Our Father in Heaven planned the coming forth of the Founding Fathers and their form of government as the necessary great prologue leading to the restoration of the gospel. Recall what our Savior Jesus Christ said nearly two thousand years ago when He visited this promised land: “For it is wisdom in the Father that they should be established in this land, and be set up as a free people by the power of the Father, that these things might come forth” (3 Ne. 21:4). America, the land of liberty, was to be the Lord’s latter-day base of operations for His restored church. - Ezra Taft Benson
You can also read God's Hand in the Founding of America.