After every reboot router is not connected to internet.
I do not change any parameters but only thru luci change order of some DNS servers,
eg. change "Use custom DNS servers 1.1.1.1, 8.8.8.8"
to "Use custom DNS servers 8.8.8.8, 1.1.1.1 "
and press "Save & apply" button.
After that internet connection works as expected.
Well, this changes everything... your router is either seriously miss-configured, or the flash memory is just about to fail. I would try to see on the kernel logs if there are more "I/O error" messages, but you should probably reflash the device and see if those errors persist.