Hamilton's 2017 title ties him with Vettel and Alain Prost as four-time winners while Manuel Fangio won five and Schumacher's seven championships still tower over the sport.
Anticipating a rowdy Mexico City reception, the Force India drivers insist they came to the race on good terms.
Verstappen thought he had a third-place finish in Texas before he was penalized for an improper last-lap pass and was taken off the podium.
Hamilton won in Mexico City last year and is chasing a fourth career season championship.
Hamilton, however, did not hesitate to admit that he "loves" drivers being able to participate in other series away from F1.
Lashing out at Connelly for his move, Verstappen said that the former is an idiot who always decides against him before also accusing the FIA of killing the sport.
Hamilton tightened his control on the F1 season title by winning the U.S. Grand Prix on Sunday ahead of title rival Sebastian Vettel, who finished second.
The Mercedes driver holds a 59-point lead over Vettel with four races to go and Vettel will be desperate to push for the win.
Hamilton has won the last three years at a Circuit of the Americas track that has favoured Mercedes' power.
Red Bull had been eager to lock down their rising star for the long term before he could be poached away.
The 36-year-old Alonso, who has 32 victories in F1, is set to enter his 17th season in racing with McLaren.
Alonso insisted that one of his main goals is to be the best driver in the world, with especially focussing on his desire to clinch the 'triple crown'.
A win in Suzuka would move Hamilton a step closer to a third F1 title in four years with Mercedes and his fourth overall.
Mercedes is 118 points clear of Ferrari and close to sealing a fourth straight constructors' championship
Having seen both of its cars fail to score points at the previous race in Singapore, Ferrari lost Kimi Raikkonen moments before this race.
Hamilton secured the 70th pole of his career in qualifying, and his fourth straight in Malaysia, while Vettel failed to set a time because of an engine problem.
Vettel, meanwhile, was handed a 20-place grid penalty after Ferrari made several component changes to his engine before the race.
Max Verstappen qualified third ahead of Red Bull teammate Daniel Ricciardo, while Hamilton's teammate Valtteri Bottas was fifth.
It was a reversal of Friday's second practice when Vettel was quickest ahead of Raikkonen.
Malaysia, however, is dropping off the calendar next year due to a massive shortfall in attendance at the Sepang circuit.
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