Squire or esquire, which are essentially the same word (the e- prefix seems to emerge as early as late latin, where some speakers had difficulty with words beginning with sc, and so they developed an esc sound instead, which often in French later dropped the s to become état from status) descend from Latin via French.
The Old French word escuier literally meaning shield bearer became the Middle French esquier, and then entered English as esquire and modern French as écuyer. It is not related to the Latin equus, meaning a horseman or knight, but to the Latin scutarius (shield bearer) from scutum, meaning shield.
Scutum is of uncertain origin. It may come from the PIE root *skewH- meaning to cover, or from the PIE root *skey- meaning to cut. The second root is the origin of the Irish scíath, meaning shield, and also the Proto-Germanic skaiþaną, which ultimately gives English shed – as in to shed skin. This root also gives the vulgar slang to shit.