Both claims are false.  While it is true that the words "separation of church and state" do not appear in the Constitution, the writings of key Founding Fathers and other documents from the period demonstrate that our government is not based on any religion, and the founders intended a separation of church and state to ensure religious freedom:

• Thomas Jefferson – Third U.S. President: Believing with you that religion is a matter which lies solely between man and his God, that he owes account to none other for his faith or his worship, that the legislative powers of government reach actions only, and not opinions, I contemplate with sovereign reverence that act of the whole American people which declared that their legislature should "make no law respecting an establishment of religion, or prohibiting the free exercise thereof," thus building a wall of separation between Church and State. (Letter to the Danbury Baptists, January 1, 1802).

• James Madison – Fourth U.S. President: [T]he number, the industry, and the morality of the Priesthood, & the devotion of the people have been manifestly increased by the total separation of the Church from the State. (Letter to Robert Walsh, March 2, 1819).

• U.S. Treaty No. 122, "The Treaty of Tripoli," Clause 11: As the government of the United States of America is not in any sense founded on the Christian religion, - as it has in itself no character of enmity against the laws, religion or tranquility of Musselmen [i.e. Muslims] … it is declared by the parties that no pretext arising from religious opinions shall ever produce an interruption of the harmony existing between the two countries. (This treaty was negotiated under George Washington's Administration; President John Adams, Second U.S. President, submitted it to the U.S. Senate; the 5^th U.S. Congress unanimously ratified the treaty; and President Adams signed it on June 5, 1797).