Was Gilligan's Island inspired by this?

Both have their charms. This one by JM Barrie has teeth.

A group of upper class Brits are stranded on a desert island with two servants. The fancy people prove to be hapless. The efficient butler begins to rule the group as a benevolent dictator.

Then they are rescued and return to high society...

The Admirable Criton was one of the most popular plays in the English speaking world for most of the 20th century. So it probably was an influence.

Just like No Exit was clearly an influence on The Good Place. (In an interview one of the writers who thought No Exit was an obscure play noted that the whole writing staff was asked to read it.)

Most of the staff of Gilligan has most likely seen a production of Crichton. Or one of the film adaptations. (Starting with Male and Female with Gloria Swanson.)

I thought it was Roger Ebert's inspiration for Beyond the Valley of the Dolls.

Just scrolling by, I thought OP was going to mention "Mary Rose" for whatever reason. The plot had potential, but the play is dreadful. It WAS fun building the bustle gown for Mrs. Otery, though. It was one of only two shows for which we built Second Bustle Era gowns, during my undergrad. The other was "Another Part Of The Forest".

Anyway, I haven't seen the film OP linked, but I'll definitely have a look.

It's worth watching. The ending was controversial :)