I downloaded and installed API Proxy on my Mac (2015 MacBook Pro running OS Sierra). I double click the program to run it. The icon appears in the tray at the top of the screen. A window appears that reads 'Loading Login Page...' then everything disappears.
According to the Proxy version, the filepath on Mac is :
or \users\<username>\Eikon API Proxy\Logs\Eikon.*
Another way to find logs with Finder : sort on date created then check last created files named Eikon.*
About the possible cause, we found that several users facing to the same issue had expired their trial account for Eikon (https://community.developers.thomsonreuters.com/questions/16551/view.html)
So you should verify your permissions for Eikon with the support.
It seems to be similar to https://community.developers.thomsonreuters.com/questions/16551/eikon-api-proxy-does-not-open.html issue except you reproduce the issue on Mac.
Could check the Eikon.<yyyymmdd>.p<iiii>.log that is generated each time you start Eikon API Proxy and share the content ?