If you go through customs as an american, and you do not waste time, you should be ok.
Business class does not help.
Remember that they will put you on next flight if you miss the first one.
Customs may well take longer unless you have GE
Of course business class helps - you'll be on top priority for standby on the next flight.
OP- get the shortest connection, if you miss it or don't feel like runing, ask a Club/helpdesk and they will gladly place you on the next flight. If there is a direct flight with (any) seat available, miss this connection and you'll be placed on the direct flight. Been there, done that.
Enjoy MSY, it's an awesome place.