Thanks to phrases like "the daily afternoon squall" common in travel writing about Southeast Asia, many people think of "squall" as simply meaning "tropical rain shower." Meteorologically, though, a squall was originally defined as wind, not rain. Here's the precise definition of both terms and how they relate.
In 1962, the World Meteorological Organization defined a squall as wind speed rising sharply by at least 8 m/s to reach at least 11 m/s, sustained for at least one minute. What sets it apart from an ordinary gust is that it lasts longer and the wind direction shifts sharply too. In other words, a squall's original definition is a wind phenomenon measured by speed and duration — not rainfall.
A shower happens when strong surface heating lifts air sharply, building a cumulonimbus cloud that releases a brief burst of heavy rain. Korea's summer afternoon showers are a classic example — rain pours down suddenly, then within tens of minutes stops as if nothing happened, with the sun returning.
In the tropics, the sun sits high and heats the surface intensely through the day, triggering an intense convective shower almost every afternoon like clockwork. These showers are frequently accompanied by sudden gusty winds (a genuine squall) from the downdraft, so the wind and rain get perceived as a set — leading popular usage to settle on calling the shower itself a "squall." Strictly speaking this is a bit of a meteorological misnomer, but it's now widely used as an established figure of speech.
What they share: both appear when the atmosphere is unstable enough to build a cumulonimbus cloud, and both start suddenly and end quickly. What differs: a shower refers to the rain itself, while a squall originally refers to the sudden wind. It's just that the tropics' daily afternoon showers often come with gusts because of how intense the convection is, so "shower plus gust" commonly appears as a set. Korea's summer showers can also bring a brief burst of strong wind when intense, but not with the same daily regularity as the tropics.
Showers can be anticipated somewhat through rain probability in the forecast, but because they're so localized, timing can vary widely hour to hour even within the same area. Saving the places you check often in sanCheck lets you check hourly rain probability alongside wind information to time your outings.