Don't know for sure, but in cases where the problem turned out to be specific to particular user account (OS user account, not Filemaker account), I've fixed problems by setting that user up with a new user account.
I am having this issue as well. I have a MacBook laptop running IS X 10.9.5 that I take back and forth with me to the office. At the office, I have no problem connecting and staying connected remotely to my FMP files that live on a "host" computer. When I try to connect remotely from home, however, I get kicked off all the time, and when I try to reconnect, often the HOSTS box on the left is empty. Strange thing is that my colleague, who exclusively accesses the files remotely from her home, never has this problem. My IT guy can't figure it out and is recommending I subscribe to the more expensive FMServer. I have a small company, with 3 users and FMP sold me this solution. Any assistance is welcome. Thank you! Lorena