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You are arriving from an international destination (Melbourne) so you will pass through customs and immigration control on arrival in the USA. There is no exception for transit passengers. You should receive both boarding passes in Melborne but if not or if you lost the second, you'll need to visit the check in desk or the baggage drop desk to get your next boarding pass. Once you have it, then you will proceed to security for your next flight.
You may need a visa or a visa waiver to enter the USA, depending on the passport you hold. (No exceptions are made for transit passengers.)
Your timing is tight but not impossible. If you miss your flight to Mexico for any reason beyond your control (immigration delays or inbound aircraft delayed), United will reaccommodate you on another flight to Mexico at their expense.