Is the game of squash named in any relation to the vegetable squash and either of them related to the action of squashing something?

Yes, they are all related—sort of.

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I’ll start at the beginning, or as close to the beginning as I can get in etymology; Indo-European.

The other day I mentioned an Indo-European root word kwet meaning “to shake.”

This word root is the same parent of the Latin word which though French gave us squash, which according to The Oxford English Dictionary means “to squeeze, press, or crush into a flat mass or pulp.”

I’m only guessing but I’d say that the violent or destructive aspects of the Indo-European kwet “to shake” morphed into the destructive aspects of squashing something.

This version of the word squash didn’t turn up in English until 1565 which is a little late for something that came from Old French with the Normans and perhaps for that reason the OED mentions that squash also might have come about in this form based on the similar word quash that had been known in English since 1275.

You squash a bug but you quash a suggestion.

In either case the result is the same, the bug or the suggestion are dead and gone.

The name of the sport squash comes from this same source. At first it was just the ball that was called the squash, the game was called squash rackets.

The ball was called the squash because it was a soft squishy ball.

I said that the vegetable squash was also etymologically related and in saying that I was stretching the truth.

A squash that you might serve for dinner or cook into some soup is named as an abbreviation of a Narragansett Indian word  asquutasquash.

In fact since most of us only eat one squash at a time the appropriate Narragansett word should be asquutasq since asquutasquash is plural.

Moreover, since we usually eat our squash cooked the appropriate Narragansett word really should be utasq since asquutasquash means plural uncooked squashes.

And how does that remotely relate to Latin and squishiness?

Before the squash ball was called squash there was an earlier English vegetable called squash.

William Shakespeare mentioned it. Back in his day the vegetable squash was immature or unripe pea pods.

The reason pea pods might be called squash was that before the peas are fully formed you can feel the emptiness of the pod by squashing it between your fingers.