If you want it to for story purposes, I guess it could... but by default? No.
They could probably just command the bead to become a fire again, even IF it went out in the first place...
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Entrap works on creatures, not items. A campfire bead isn't extinguished by cold. So, RAW, the breath weapon will do nothing to it.
Honestly, I feel that the ability would benefit from a bit of descriptive text. We can assume that the thremidyr entraps with a layer of ice and that the ice clings to objects too, but RAW it entraps with a non-descripted method and the entrapment operate only on creatures.
For story purpose saying that the fire is encased in ice (if the item fails its saving throw) until it melts it sounds good, having the group unattended equipment and meal enchased in ice too sounds good. I imagine the scene of some film on artic expeditions or people searching for gold in the Klondike.