All teams have stars-and-stripes caps for 7/4, etc.
Yes, but they still show the Team logo. Back after 9/11 the Mets came out to play, first game of pro sports back in NYC, and came out with all different NYC fire and police department hats on. The league issued a warning saying not to wear them or they would be fined heavily or not allowed to play (honestly I can't remember) and the Mets came out with them on anyway (F Yeah, 'Merica!) but changed them right before opening pitch.
It was a great tribute that was squashed for a stupid rule. There should have been a write in addendum that allows for all teams to wear league approved (Basic base colors like black or navy, nothing bright green or orange unless you play in South Beach) hats without the team logo, but "ladder co. XX, NYFD" on the front.