Born in Budapest, Megyeri began his career with Ferencvárosi TC and played a short time in the youth side for Bristol City, before returning to his previous club in 2008.

Megyeri made his debut for the club on 3 April 2011, playing the full 90 minutes in a 3–1 home win against Kavala.

After Nikopolidis' retirement and Pardo's departure to APOEL, Megyeri was challenged by new signings Franco Costanzo and Iosif Daskalakis.

After Constanzo's poor performances, he was elected as first-choice; despite the arrival of Roy Carroll in January 2012, he continued to appear regularly, contributing with 22 matches.

On 15 July 2015 Megyeri signed a three-year deal with La Liga side Getafe CF, after his contract with Olympiacos expired.