Wheel bearings, or hub bearing in the brake drum, or deteriorated swing arm rubber bushings, might cause it.I'd pull the rear wheel and have a look inside there to see what's what.
His bike should have bronze bushings and a tube, no rubber bushings. ERC
Well, in that case, he should check the bronze bushings for wear.
My G5 Deluxe felt very squishy(as you describe) at the rear UNTIL I GOT RID OF THE AVON SKIDMASTERS, got better shocks also & Dunlop K70's.