Yes, you can wear them ANY time. If you buy a good brand, and one that fits, they provide excellent support. I'd certainly never try to go running in one of my every day underwire bras...but my 'high impact' sports bras support my breasts when running so I have no discomfort (and I was out running 3 days after having surgery on one of my breasts).
Your friend apparently knows nothing about sports bras.
Now, one issue you may find is that a lot of sports bras provide more coverage of your breasts, back, and wider straps than non-sports bras but if that isn't an issue for you, then wear them whenver you like.
After my breast biopsy, I was told by the medical staff to wear a sports bra for support and comfort, not a regular bra (until the wound healed). I also often sleep in a sports bra and wear them just around the house for support and comfort...underwires aren't comfy to lay around in.
As for "will they make them full"? I've no idea what you are talking about. Will what make what full? Sports bras are supposed to be snug, but not uncomfortable. They don't change the size of your breasts but can make it appear that you have less than you do, because a sports bra isn't designed to enhance but to hold them close to your body and support them.