Does his ability work on himself?

Yes.

Rhymer is friendly to himself, and also at distance 1 of himself.

Note that his rule says "Friendly squadrons" rather than "Other friendly squadrons" like Howlrunner, who does not apply to herself.

From the FAQ:

Major Rhymer

Friendly squadrons that are at distance 1 of Major Rhymer can attack ships at close–medium range instead of being restricted to distance 1. This includes Major Rhymer himself.

And, more generally, from the Rules Reference, p.6:

Friendly and Enemy

A ship or squadron is friendly to itself and can be targeted by an effect that specifies a friendly target unless it explicitly specifies “another” ship or squadron.