What worries me is that you cannot uncompress the installer file in the File manager.
You will have a problem in a daily operation if you cannot uncompress file.
Can you open File Roller and try to uncompress file from there?
Or try uncompressing online from a website.
I suggest you to chage the name of the installer something simple such as
brother (it will still run I tested it myself)
Open terminal
cd Downloads
sudo bash brother
CUI will ask you
Input model name ->
Type the printer name exactly as follows:
MFC-J5620DW
Follow the instruction thereafter.
We're getting somewhere.
But I still can't scan after rebooting both computer and printer.
The printer is connected by WiFi, the PC by LAN
How are you connecting the printer?
USB or WiFi?
If it is WiFi then you need to connect the printer to your local network.
Assuming your printer got a similar control panel as mine, go to set up then press WiFi assistance (mine is in French, English wording may be different).
With Win10 the scanner worked using WiFi to the same network connected to the PC by a LAN connection.
Guess I'll have to run out and buy a LAN cable.
I'll let you all know if that solves the problem
In the meantime, thank you for all of the assistance.
Cheers 
You cannot connect printer with LAN cable to the computer.
You need to connect it to the router.
Is there anyreason you do not want to WiFi connection in Linux?
I know -- I meant the LAN goes to the router.
I would prefer using WiFi and not having to run a cable across the room.
I ran the WLAN assistant and it's connected, but the scanner still doesn't work.
I've got to step away and run some errands.
Thanks for the help again.
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Ah, OK.
The way you wrote suggested otherwise 
It sounds like we will have some more fun later then 
This might help:
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That worked
I was able to scan finally!
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