Ted Cruz is disqualified to be president and not just because he's a flaming asshole and a traitor or because Trump says so. Ted Cruz is not a natural born citizen.
Section 1 of Article Two of the United States Constitution sets forth the eligibility requirements for serving as president of the United States, under clause 5:
The Twelfth Amendment states, "No person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States." The Fourteenth Amendment does not use the phrase natural-born citizen. It does provide, "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside."
Under Article One of the United States Constitution, representatives and senators are required to be U.S. citizens, but there is no requirement that they be natural born.