We renovated our house last year and now we need to add some more lighting that we didn’t think we would need at the time, so…what we have.  We had fans added in two locations and we want (4) can lights added around each, electrician ran 4 wire (white, black, red, ground) to each fan box connected to 2 switches (one for fan, other for light), however our fans are controlled by a remote so the light switch is capped off in the fan box.  I’m comfortable working with electric, but not comfortable enough to figure this out.  My thinking is to hook up the red line from the fan box to the black on the new can light and pig tail the white and ground.  My thinking correct?

If the red is not being used and is from a switch, yes.
What you are describing is correct.

It's great if your electrician ran a 2nd wire, most don't.

If it's not being used as a traveler for a 3-way switch or something then yes, that's a solid plan.

Sounds like the black ungrounded conductor is being used to power the remote in the fan box and the wall switch controls power to the remote. Since you added the remote, the second switch and the red wire are no longer used for the fan. If that's a correct interpretation, then yes, you can reconnect the red to the second switch and then the red from the fan box will connect to the black for the can lights. The can lights white will connect to the white in the fan box, same for the bare copper ground.

Doesn't that make the neutral wire carry double it's rated capacity?
You end up with two 120V circuits.  Black white and red white.
The red and black are 180 degrees out of phase.
But isn't the white carrying the combined amperages of both circuits?

If you ran a separate circuit from each side of the CB panel, they'd both come with their own neutrals.

Agreed on the above.

Ahh yes, it is most likely a 2-wire feed to the switch box rather than a multi-wire branch circuit.

Upon reflection, I'm guessing it's wired to the ceiling box in the same manner as a 3 way switch, using the 12/3 as a local traveler.
Now that I think of it, that would make the red wire the same as the black, if they are both connected to the 12/2 home run at the wall box.

Which would make this ok as long as it was all treated as a single circuit for capacity calcs.

I say "phases" because it's not a 2 phase system, it's a split phase system.  But the again, I'm splitting hairs...

When he installed the fan box he ran 3 wire to it and told me if the fan we purchase is prewired for a remote that I was to cap off the red wire.  As mentioned two switches were installed, one for fan and the other for the light, he informed me if we get the prewired fan that the light switch will be "dead".

The only pertinent thing I think should be added to this thread is that on a circuit using two wires originating from different sides of the panel (ie, 240 between them) and sharing a neutral, only the unbalanced load between the two returns on the neutral (white).  In other words, if they are both running at 15 amps, zero current would be on the neutral.