No. You can still get Zodiac Spear even if you have open those chest.
Chances are lower though.
Here is how:
Location 2 - Henne Mines, East hidden section.
- Go North from the Gate Crystal, then East.
- In the secret section, turn right from the second junction. Follow
the way until you reach to next mine route. Now pay attention
to right side. Around crates, there is a small area which didn't
shown in the map. The treasure chest will appear there.
- Only 10% chance you can find the treasure, and only 10% chance
could get this weapon.
- Here is the point
* Requirement - Must have at least 10 espers and finish Mind Flare mob hunt
mission (rank B).
- Go Land Of The Garif find the geomancer to get some info, he's
located at opposite of the Item Shop. Now back to Henne Mines,
the gate to East has been unlocked.
- Equip Diamond Bangle.
*Credit goes to sephirosuy from GameFAQs.