Odds are that you'll start in Texas. I think a citizen of Texas would be preferred, but from what I hear, they accept USA citizens as well.
In other oil patches within the US - Gulf Coast, North Dakota, San Joaquin Valley, your mileage will vary. If I were looking at resumes, I would not be overly concerned about a Hungarian BSME. I think most folks would have the same attitude - but certainly not all. As others above have pointed out, international life experience is a valued commodity. Use that as an advantage, don't worry about it as a weakness.