Akpan, 19, made his only appearance for the Toffees as a substitute in the Europa League against FC BATE in 2009.
He spent time on loan at Hull City in the 2010/11 season, making two appearances for the Championship side.
Crawley manager Steve Evans told the club website: "He had many offers from Football League clubs to consider but I have kept in regular touch with him."
Evans continued: ''I watched him play several times last season for Everton reserves, indeed I tried to get him on loan at that time, but David Moyes said no to me and a few other interested managers.
"I made sure we monitored him thereafter and the report from his performance when he played for Hull City against Queens Park Rangers made my decision very clear, we should try and sign him if we could.
''Being honest I did not really fancy my chances of getting the lad here, but if you don't try then you would never know was my thought process, and we have got him.
''He is a big lad who can be aggressive and strong, but he is a footballer. He has a good range of passing and a terrific work ethic, he is young, and I think he can become a big player for us."