How can I compare Jay Gatsby of The Great Gatsby to Allie Fox in The Mosquito Coast?

That's an interesting comparison and one that seems to lend itself to more differences than similarities, so I'll start with some contrasts.
The American Dream: While Jay Gatsby seeks to achieve the idealistic American Dream at any cost, Allie Fox is determined to leave it all behind at any cost. Jay Gatsby has one goal: Daisy. And to win her love, he amasses a great fortune of both money and property (likely through illegal means). He pours his wealth into his parties and has an open invite every weekend to showcase his grandeur. On the other hand, Allie Fox leaves America and its consumerism for a remote village on the coast of Honduras. He summarizes his feelings about the American excess here:

We eat when we’re not hungry, drink when we’re not thirsty, buy what we don’t need, and throw away everything that’s useful. Don’t sell a man what he wants—sell him what he doesn’t want.

Jay Gatsby represents everything about America that Allie Fox detests.



Reliance on others: Jay Gatsby has a need to be needed. He throws lavish parties and befriends Nick, confiding in him that he was in love with Daisy. He longs for her and has planted himself in New York—in the center of activity and abundant opportunities for social events. Conversely, Allie Fox not only leaves America for the more remote Honduras, but after the explosion that destroys Jeronimo, he takes his family even farther through the jungle into the more remote Laguna Miskita. He keeps seeking further isolation as the plot develops.

However, the men are also similar in one major regard.

Their goals kill them: Jay Gatsby puts everything on the line to win Daisy's favor. Allie Fox puts everything on the line to prove that American technology and consumerism are corrupt. Both men follow these paths to their eventual deaths, both murdered by gunfire. Thus, neither man achieves the dream that he so desperately sought.

Both Jay Gatsby and Allie Fox perish because they try to hold onto a distorted version of the American Dream. Although Jay has been extraordinarily successful on the material level, growing fantastically rich, he is not contented within himself because he does not have love. Specifically, he wants the love of his dream girl, Daisy. Gatsby and Fox are both romantics in that they cannot confront the humdrum reality of ordinary existence, but believe that their own reality must hold something more. Gatsby's downfall comes by way of Daisy, who sets up the circumstances that finally cause Wilson to kill him. Fox's dreams are more dangerous because he hauls his family down to Central American on the basis of a whim. As a father, he has more responsibility than the single Gatsby, but he refuses to acknowledge that responsibility. Ultimately, Fox causes his own demise because he is blinded by arrogance. His American Dream is that of expansionist imperialism overseas, while Gatsby's was more confined to capitalist entrepreneurship (albeit illegal) at home.