I have no idea whether hot glue would work, but I sort of doubt that any nail product would produce a very durable repair.
You might try one of the spray matte fixatives used by artists that a lot of fly tiers use to strengthen duck/goose/turkey quills.
Or..
There is this stuff called Aleene's Fabric Fusion that you can get a 20 year supply at Wal-Mart for 5 bucks or so. I use it as a substitute for that way-too-pricey UV stuff that is used to coat eyes and make clear heads on big bass and pike flies. You want the water based variety of Fabric Fusion. Use water to thin it down to the viscosity you desire and coat your feathers very lightly with it. Then they need to dry at least overnight. Fabric Fusion dries clear and despite being water-based, the final product is waterproof and pliable.
