[Yosemitest]

"Cruz Sr was either a legal immigrant to Canada or a full-blown Canadian citizen at the time.
The Cruzes haven’t clarified exactly when Cruz Sr got his Candian citizenship. "

Source from JUNE 20, 2013 for answer tho your question

.. 74-Year-old Rafael Bienvenido Cruz:

"I came to this country legally," Cruz's father says.
"I came here with a legal visa, and ... every step of the way, I have been here legally."

In an interview near his home outside Dallas, the elder Cruz says that as a teenager, he fought alongside Fidel Castro's forces to overthrow Cuba's U.S.-backed dictator, Fulgencio Batista.
He was caught by Batista's forces, he says, and jailed and beaten before being released.
It was 1957, and Cruz decided to get out of Cuba by applying to the University of Texas.
Upon being admitted, he adds, he got a four-year student visa at the U.S. Consulate in Havana.

"Then the only other thing that I needed was an exit permit from the Batista government," Cruz recalls.
"A friend of the family, a lawyer friend of my father, basically bribed a Batista official to stamp my passport with an exit permit."

The Rafael Cruz that his son Ted portrays is a kind of Cuban Horatio Alger
— arriving in the U.S. with only $100, learning English on his own and washing dishes seven days a week for 50 cents an hour.

"Since he liked to eat seven days a week, he worked seven days a week, and he paid his way through the University of Texas," Ted Cruz says
of his father, "and then ended up getting a job and eventually going on to start a small business and to work towards the American dream."

Only he did that in Canada, where Ted was born.
His father went there after having earlier obtained political asylum in the U.S. when his student visa ran out.
He then got a green card, he says, and married Ted's mother, an American citizen.
The two of them moved to Canada to work in the oil industry.

"I worked in Canada for eight years," Rafael Cruz says. "And while I was in Canada, I became a Canadian citizen."

The elder Cruz says he renounced his Canadian citizenship when he finally became a U.S. citizen in 2005 —
48 years after leaving Cuba.
Why did he take so long to do it?
"I don't know. I guess laziness, or — I don't know," he says.

Peter Spiro, a legal expert on U.S. citizenship at Temple University, says Rafael Cruz followed "sort of a zigzag path to citizenship."
Spiro says Cruz's multicountry odyssey did not follow traditional models for immigration.

[Nero Germanicus]

When the Treaty of Paris of 1898 was signed which ended the Spanish-American War, the U.S. received Puerto Rico, Guam and the Philippines as territories. Under the peace treaty Spain also gave up control of Cuba and the U.S. administered the island from April, 1899 until Cuba became the independent Republic of Cuba on May 20, 1902.
Cuba was never an official U.S. Territory.

[Plummz]

No answer there.