Can FC Connect run on a Linux server?
No. ABBYY FlexiCapture Connect can be run only as a Windows service. More information about FC Connect is here: https://help.abbyy.com/en-us/flexicapture/12/cloud/cloud_fc_connect/
But for the import and export on Linux, you can use FC Cloud API. More information about it is here: https://help.abbyy.com/en-us/flexicapture/12/developer/using/