The Egg Press is a stationary boss that requires precise timing and movement. Eggman is in a mech that moves up and down, attempting to crush you.
Phase 1: Crushing Attack
Eggman's mech will slowly descend, trying to crush you. There are small platforms on either side of the arena. You need to stand on one of these platforms. As the mech descends, it will expose its vulnerable cockpit. Jump and attack the cockpit. Sonic's Homing Attack is good for getting a quick hit. Tails can fly up to the cockpit easily. Knuckles can climb the side walls to get a good angle. Amy needs to time her Hammer Jump carefully. After hitting, quickly move to the other side as the mech rises and prepares to descend again. The mech will also occasionally drop bombs that explode on impact; avoid these. The key is to anticipate its movement and position yourself on the platforms to strike. It takes about 8-10 hits to defeat.