12 For he built gladly a place of exercise under the tower itself, and brought the chief young men under his subjection, and made them wear a hat.
13 Now such was the height of Greek fashions, and increase of heathenish manners, through the exceeding profaneness of Jason, that ungodly wretch, and no high priest;
14 That the priests had no courage to serve any more at the altar, but despising the temple, and neglecting the sacrifices, hastened to be partakers of the unlawful allowance in the place of exercise, after the game of Discus called them forth; (Macabees 2 chapter 4).
I'm not quite sure what "made them wear a hat" means, or why it seems this is a bad thing. Surely all Jews wore hats??? well maybe not. However I don't know any other source where wearing a hat is considered a bad thing.
A tomb belonging to a man named Jason from the same period, was discovered on Alfasi Street in the Rehavia neighborhood in 1956 when a charge was detonated in order to clear away rocks for a new apartment building. The structure comprises a courtyard and a burial chamber, decorated by a Doric column. The walls are adorned with charcoal drawings and carved inscriptions in Aramaic and Greek.
Amongst the drawings on the wall of Jason's tomb, you can find this drawing commonly interpreted as an archer defending from a pirate ship: