I climbed Squaretop with my friends and family over labor day weekend in 2019 It is a pleasant hike pretty much the first ten miles to beaver park.
From Pinedale Continue west onto US-191 then in 6 miles turn right onto WY-352 W then continue strait 44 miles to green river lakes trailhead.
Once you are at green river lakes trailhead walk the highline trail for 11 miles until you reach beaver park, when you reach Beaver Park many people stop for the night and camp there before they make there accent. From Beaver Park walk the left edge of the meadow north and you will find a trail follow that trail up 1500 ft elevation gain to Granite lake, from there you will take a left when the trail merges then bushwack up to a bench from there the trail should be visible in the hillside when you find the trail going up keep on it until you reach the saddle which is about 1500 ft higher then granite lake. When you reach the saddle follow the trail around a rock ledge after you get above the rock ledge you will see why its called Squaretop because its just flat on top. continue north until you reach the highest point and then you've made it.
There are no permits needed for climbing Squaretop.
From July 1st to Sep 10th is the best time to climb because all the snow is pretty much melted.