There are no Linux download links. The download page just shows image placeholders.
But if you send an invalid download request like ".../download?platform=foo" then you get back a handy JSON error string indicating that the Linux downloads are:
https://updates.getmailspring.com/download?platform=linuxDeb and https://updates.getmailspring.com/download?platform=linuxRpm
Maybe Ben or someone will see this and comment why the links aren't there, since it looks like the github repo has issues filed for this.