In Pathfinder the rules for squeezing are as follows:
In some cases, you may have to squeeze into or through an area that isn't as wide as the space you take up. You can squeeze through or into a space that is at least half as wide as your normal space. Each move into or through a narrow space counts as if it were 2 squares, and while squeezed in a narrow space, you take a –4 penalty on attack rolls and a –4 penalty to AC.
When a Large creature (which normally takes up 4 squares) squeezes into a space that's 1 square wide, the creature's miniature figure occupies 2 squares, centered on the line between the 2 squares. For a bigger creature, center the creature likewise in the area it squeezes into.
A creature can squeeze past a creature while moving but it can't end its movement in an occupied square.
To squeeze through or into a space less than half your space's width, you must use the Escape Artist skill. You can't attack while using Escape Artist to squeeze through or into a narrow space, you take a –4 penalty to AC, and you lose any Dexterity bonus to AC. [Emphasis mine.]
By the Rules: Creatures Squeezing
If the DM determines that the space the horse needs to occupy is half or more its space but less than the space it usually occupies, the horse can squeeze like any other creature. Having the horse squeeze is just riding it or leading it; any creature can squeeze, and no particular skill or training is required to do so.
House Rules: Squeezing Horses
It's a house rule that the creature's owner must make Handle Animal checks to permit the horse to squeeze. Depending on the campaign's treatment of horses (e.g. horses are vicious creatures who pursue foes into burrows all the time, horses are cowards who won't enter areas in which they can't fully maneuver), the Handle Animal skill check DC could be 10 or could be 25; it's totally the DM's call.
The following is for completeness in case the DM rules the horse must fit into a tight space and not merely squeeze.
Creatures Moving into or through Tight Spaces
If the DM determines that the space the horse needs to occupy is less than half the space it usually occupies, then the horse must use Escape Artist checks to (even briefly) occupy that space. Having the horse occupy doesn't require any special Handle Animal or Ride checks but requires Escape Artist checks on the horse's part.
Assuming standard horse or variant thereof, the tunnels must be less than 5 ft. wide to force the horse to make Escape Artist checks. Then, seriously, the skill Escape Artist says, "Squeezing through a tight space takes at least 1 minute, maybe longer, depending on how long the space is," and lists the DC for the Escape Artist skill check to do so as an astounding 30.
As there's no way the typical Dexterity 14 horse, even with a generous DM allowing its owner to push the horse to perform the trick Break Out for a +4 circumstance bonus to Escape Artist checks, is ever going to make a DC 30 Escape Artist skill check by itself, the horse must rely on assistance from the judicious applications of the action aid another ("C'mon, horsey! You can do it! A little to the left! Squeeze!").