Text of House Clegane Brigands is "If you are running a character agenda or are not running an agenda, House Clegane Brigands gains: 'Response: After you win an challenge, stand House Clegane Brigands'."

If I win an int challenge, can i stand HCB while my opponent revealing Burning Bridges?

No. The stand response is a triggered effect.

Yes, you can.

The triggered effect is a gained ability, and is not affected by Burning Bridges.

Oh - yes, blasted self-referential gaining... that twists me up every time.

sWhiteboy is correct, as the triggered ability is not "on" the card, but gained by the card (despite it being gained by itself), it is not affected by Burning Bridges.

Another popular such case is "Maester Murenmure".

And the Robb Stark that reduces armies!

And The Iron Throne (LotR).