It Is only recently that a passport is required to re enter the US from Mexico and Canada.
True, but that new requirement applies to both naturalized and natural born citizens. In other words, all citizens are required to provide proof of identity that is acceptable to the government (passport) in order to enter into the US, if such admittance is to be legal.
Being a NBC does not excuse one from that requirement.
IIRC, Cruz and his wife traveled to somewhere in Central America to dig water wells. He would have needed a passport for that travel.