With some restrictions, the grappled creature can make a full attack
In the description of the combat maneuver Grapple, the section If You Are Grappled says that
Instead of attempting to break or reverse the grapple, you can take any action that doesn’t require two hands to perform, such as cast a spell or make an attack or full attack with a light or one-handed weapon against any creature within your reach, including the creature that is grappling you.
Natural weapons are considered light weapons. The eidolon suffers a −2 penalty on attack rolls with each natural weapon attack due to the condition grappled, but as claws are primary natural attacks, it should suffer no additional penalties.
The grappled eidolon isn't taking an action to try to use its claws to employ bolt cutters, tie its shoes, or play guitar, and, instead, taking an action to use its claws to just rip up some fool grappling it. That's not a two-handed-or-more task but a task for each individual claw.