First off, Frustrated, you would only be kicked out of the US if you committed a crime on an H-1B. If the insurance company decided to charge you with felony fraud, then yeah, probably you would not be back any time soon.
Second, Geico is the one of the most unethical companies anywhere to deal with. Car And Driver magazine did an expose where Geico funded a laser radar (aka LIDAR) company called 20/20 with $950,000 of funding to finish development of their speed gun. Then, Geico bought a bunch of them at cost and handed them out to police departments across the country FOR FREE if they would write tickets with them on a daily basis. Their motive? Since they're the biggest insurer in the country, on average they knew they would ding a lot of their insured drivers with tickets and jack their premiums up. That fact ALONE should make someone drop their insurance with them. Their interest is not to support their insured, but to make money. Their rates and all the advertising they do should be a sign of that.
Third, are you moving to New Jersey? That seems like an awfully high insurance rate. New Jersey and a few other states are notorious for high insurance rates. If you are planning to work in one of those states, try to live in a nearby state (e.g. Pennsylvania vs. NJ). The rate you pay depends a lot on your nine-digit zip code. For example, my ex-manager in San Jose fought with his insurer because, although his neighborhood was very low crime, it was in the same zip code as a very bad east San Jose ghetto.
The determinants of insurance rates are:
1. Criminal convictions, esp. for drunk driving or criminally negligent driving (any conviction = more $).
2. Number of years as a licensed driver (less years = more $).
3. Number of at-fault accidents claimed against you in the last 3 years (more accidents = more $).
4. Number of moving violations in the last 3 years (more tickets = more $).
5. Age (younger than 25 = more $).
6. Marital status (not married = more $).
7. Zip code (high crime area zip codes = more $).
8. Credit rating (derogatory credit or no credit history = more $) and ONLY in some cases/places.
9. Type of vehicle (high theft rate vehicles or sports cars or SUVs = more $).
10. Amount of public liability carried (more public liability coverage = more $).
11. Number of insured vehicles (less vehicles = more $) but you may only need one policy.
12. Multiple policy discounts for home and other insurance (less policies with same company = more $).
13. Deductible amounts (lower deductible = more $).
14. Collision coverage on your own car for accidents that are your fault (collision coverage = more $).
15. Loss of use coverage on your vehicle to cover rental cars (Loss-of-use coverage = more $).
16. Your sex (male = more $).
17. Other features of the insurance policy (generally, the more benefits you choose to cover, the more $).
I'm sure I've missed a few points and repeated a few points, but these are the big ones I can think of.
One last thing - when you go to open your bank account here, make sure you go with a credit union if you can, and particularly any bank that your employer recommends or sponsors membership at. When you do, bring a copy of your credit report from your home country IF IT HAS NO NEGATIVE ENTRIES and INSIST that it be used to determine the percentage rate of a car loan. It can be difficult to even open another account if you have no credit history, much less get a car loan. Also, bring your credit report to your insurer but DO NOT disclose it unless they ask for it.
Good luck!