I've taken a fly box in my carry on before. It was an accident and when I realized it I thought I was going to lose them. I don't know they were supposed to let me through with them but I got through no problem. I think you will be ok. I worked in a knife shop so I had to be up to date with the laws. Normally anything small like flies or hooks are allowed. As long as you dont plan on bringing 15/0 tuna hooks you should be ok.
Tackle Equipment - Fishing equipment should be placed in your checked baggage. Some tackle equipment can be considered sharp and dangerous. Expensive reels or fragile tackle such as fly's should be packed in your carry-on baggage.
