Since Rice considers “state residency” in their admission decisions, they likely are trying to recruit a geographically diverse group of students (for those coming from outside of Texas). So I’d say that being from NH couldn’t hurt and could only help, though any advantage is likely to be very small.
See more details about geographic distribution here:
https://www.collegefactual.com/colleges/rice-university/student-life/diversity/#chart-geographic-diversity
Good luck!