Liz, they'll be OK a time or two but if you plan to do it daily for many days, they won't last.
Believe me, during our last trip we spend $3 in a bottle of Powerade and thought about just washing it and reusing it to save. After a few days, the thin plastic was so bent and cracked by the ice that it leaked.
LLBean is selling two of those good Nalgene bottles now for $16.50. They are nice size bottles too, 1 liter each. I am buying those for our new trip. They freeze and thaw wery well and even the cap is made of a sturdy plastic. With their unbeatable lifetime warranty, those bottles should last you forever.
Wanda