Normally an adult bird will pant like that if it has been flying hard and is stressed, needing water, or too hot. Obviously, being a youngster it hasnt been flying so that leaves stress, water or too hot.
You said you had a hot water bottle, is the bird able to get OFF the heat if it needs, also, is the bottle covered with a towel ?
maybe its a combination of all three, if its too hot & all the squeaking (which all young birds do most of the time) it is stressing as it is looking to you as its parent. Try dipping its beak in water as suggested (may have to do it several times).
Definately sounds to me like its hot and needs water.
Whatever you do, Do NOT try to force water down its throat or it may aspirate & die.
If it wont drink on its own you can dribble a few drops along the edge of its beak.