I did setup my printer successfully on 10.04, 12.04, 14.04 and 16.04, there were various troubles, but I still could manage to resolve it in the end.
It's 18.04 (64 bit, after upgrading from 16.04 64 bit) where I don't seem to be able to figure it out. Admittedly, the scanner part works for me, but not printer.
The printer is present and recognized. When it is off, the system correctly marks jobs as "waiting for printer to be available". However, if printer is on and connected, it accepts jobs to only dismiss them as "completed" jobs from the user "Withheld" (I googled it, it's CUPS thing and it is fine) the very next moment. No actual printing happens, test page or not.
The printer actually worked for me on 16.04 before upgrading to 18.04, so I don't believe the device is to blame.
So. What I've tried so far (besides re-installing CUPS, rebooting and reconnecting the USB cable).
The official Brother way. Went to MFC-7420 Downloads, got the installer script which installed the LPR and CUPS Wrapper drivers for MFC-7420 (both of i386 architecture) and scanner drivers (as I said above, the scanner part works with some hints from Internet).
The less official way. There are amd64 packages that exactly suit (well, in theory) my MFC-7420:
$ dpkg -l | grep brother
ii brother-cups-wrapper-common 1.0.0-10-0ubuntu7 amd64 Common files for Brother cups wrapper packages
ii brother-cups-wrapper-laser 2.0.1-2-0ubuntu7 amd64 Cups Wrapper drivers for laser brother printers
ii brother-lpr-drivers-common 1.0.0-4-0ubuntu3 amd64 Common files for brother-lpr-drivers packages
ii brother-lpr-drivers-laser 2.0.1-3-0ubuntu5 amd64 LPR drivers for laser brother printers
- Switched between (recommended) "MFC-7450 BR-Script3" and (apparently installed with packages above) "MFC-7420 for CUPS" drivers.
- Completely removed any MFC-7420-related package or driver to start over from the blank.
- Googled a lot, but then mostly it all points back to the Brother site which is p.1.
So, the problem is still on. I want to print, but the printer just doesn't. Any insights?