"Caught in the rush" needs some context to be fully understood. Rush (noun) means the act of hurrying. So if there was a rush for the door, that means everyone ran for the door to get out. You could be caught in the rush to the door. Rushing water is fast moving water. If you were caught in the rush of the river, it could carry you away. To give up means to surrender, to abandon or admit defeat. If you refuse to give up, you will not surrender, you decide not to accept the present outcome. Alternate ways to phrase, "I refuse to give up." I will not surrender! I won't accept failure. You will often hear, "Never give up!" which encourages one to never stop trying to accomplish one's goals.