Can you go to the public gym in your OCPs and work out wearing your undershirt, trousers and boots?

Why in God's name would you purposely work out in OCPs? The only time you do that is if you don't have time to change. You'll see it on base when people only have 30 minutes to hit the gym over lunch, but anywhere else is ridiculous.

Oh look, Im a soldier, thank me for my service sounds like what you are going for

Public gym? As in, the gym on base? If so, the gym on base will dictate what the appropriate attire is. But, I have seen it done that way.