Can I say "don't be so surprised" instead?
Yes, you can. "Don't have a heart attack" can be replaced by "don't be too surprised" without changing the meaning of the sentence, although the effect of exaggeration is lessened. "Don't have a heart attack" is a casual expression that can be used to show off your fluency in English, so using it is recommended. Ex: I have a surprise for you! Don't have a heart attack. = I have something for you! Don't be too surprised.