I live in a state that does not participate in the lottery (Powerball, Mega, etc). If I drive over to another neighboring state that does participate, buy a lottery ticket, and win, is there anything that will prevent me from receiving the prize?
No. You can still receive the prize if you are a U.S. citizen.
Lotteries are almost always run by state governments, and the privileges and immunities clauses of the United States Constitution prohibit discrimination based upon state citizenship in most cases.
There are exceptions to this general rule, for example, for hunting and fishing licenses and state college tuition, but there is no exception whose logic would apply to lottery tickets payable in money. (One could imagine an exception if lottery tickets, for example, were payable in marijuana.)