OK. Well these are the results.
I uninstalled Microsoft Security Essentials, disabled firewall and installed BBP version 0.90.69 again – now it works. So, at least in Windows 7 (64 bit) the issue is Security. If you have Windows firewall and Security Essentials ON – BBP won't work.
ALSO: I installed newest version (demo) 0.90.95 to my HP PC, it has Windows 10 Pro 64 bit, insider build 14352 with Windows Defender and Firewall ON – Breakaway installs and works just great! I hope you guys at Claesson Edwards are aware of this. So at some point BBP will work with Windows 10 when the W10 update comes to "normal customers".