Network connection is fine here. Not being able to check for updates maybe indicates a firewall issue.

Is this windows? If so windows firewall may be blocking the Espers-qt.exe application. For a possible quick fix exit Espers and try moving Espers-qt.exe to another folder, launch Espers-qt.exe and give windows firewall permission for private and public networks (windows may be treating yours as public blocking Espers now)