When using Winbox how does one setup a basic firewall rule? What is the equivalent to doing the following via Ubuntu's UFW (Uncomplicated Firewall) using the following example:

# Deny all incoming traffic by default.
sudo ufw default deny incoming
# Allow printing via IPP (Internet Printing Protocol) on TCP & UDP.
sudo ufw allow 631

It really depends on what level of security you want to achieve. When blocking all incoming connections, one should be really careful to specify the incoming interface, otherwise you would be left out of the router.
Yoy may have a look at:
http://wiki.mikrotik.com/wiki/Securing_your_router
http://wiki.mikrotik.com/wiki/Port_Knocking
http://wiki.mikrotik.com/wiki/Bruteforc ... prevention

What I am after is how to add a firewall rule in Winbox that will allow certain traffic to go through. I have a client that is desperately trying to print, but is forced to connect to a different network (via the Linksys router which has a firewall which is reasonably straightforward to setup) in order to do their printing.

Post your firewall filter to have a look what is causing the block. Normally the firewall filter is blank, which means nothing is blocked through it, unless you have preserved the default configuration, or you have added some of your own.
From where to where is the person trying to print? Are the two networks connected through the router? Maybe you are looking for NAT configuration instead for filters?