I was in a similar situation, and have a life-long history of panic attacks. I would have them during stand-ups, which was absolute hell. I went on to face this, becoming a public speaker (technically speaking) and also a professional skydiver and occasional BASE jumper.
Things that help me in general are cardio, weightlifting, mindfulness exercises and meditation. Knowing I can walk away from a situation and be ok financially is huge.
Things that helped me specifically with daily stand-ups:
1. Preparing questions. Asking a probing question puts the pressure on the person that you feel is evaluating you. If you have a blocker, begin with asking the other engineer if they have completed that dependency. Always have a question in the chamber. Be careful not to throw others under the bus, especially if they are anxious or junior. The best questions are directed at the higher ups. After a year I learned that the executives were nervous about my questions and joked about my reputation as a "hard-hitter". That really helped me to feel more in control. Never put the company in a bad light with your phrasing.
2. Volunteer. Being third in line is panic inducing. Combat this by jumping the gun. I was often shocked to hear myself say "I'll go first" or "I'll start". My heart rate would jump a little but my brain didn't even have time to sort out my self-trickery before. The relief you feel after doing this is incredible. Also, there's something inherently calming about deciding to speak, rather than being expected to.
3. Conversational style. Providing an update can be terrifying. Having a casual conversation is [hopefully] not. Make it a back-and-forth. Not all of your questions need to be interrogations, simply passing the buck with a softball like "Can you clarify..." or "I'm thinking of doing this, what's your opinion?" make all the difference.