Nothing would happen. Hawaii sent out an alert to every single cellphone in the state that an ICBM was imminently approaching Honolulu a few years back, and no one outside of Hawaii remembers that anymore today.
We lived on O'ahu when it happened and the topic still comes up occasionally. At the time, it was quite traumatic for myself, my wife, and our kids. I spoke to a large number of people who were still in a state of shock in the following week. At least one person died of a heart attack.
Errant tweets or text messages like this from positions of authority have the potential to have very real and major effects on people's lives. Don't downplay the power they hold.
People who think they're "good at evaluating information" and ignore emergency notifications can and do quite often end up dead.
The core rule for any alerting system is to trigger if and only if those-to-be-alerted must take some sort of action. In this case, I was tired, pardon the pun, of being blasted out of a deep slumber at 3am for an earthquake that was so weak I could not feel it.
The only action I needed to take was to try and fall asleep again, which isn't terribly easy after your phone has jacked itself to maximum volume with urgent news of impending doom.
After the fourth time in something like two weeks, that alarm was disabled. It was useless.
Kudos to the designer for the earthquake alarm sound, though. That bit does its job brilliantly -- it instantly grabs your attention without deafening you, and you know exactly what is going on.
BALLISTIC MISSILE THREAT INBOUND TO HAWAII. SEEK IMMEDIATE SHELTER. THIS IS NOT A DRILL.
I turned on the radio and FM stations were playing the Emergency Broadcast System's well known emergency tone followed by a computerized voice playing the same message as above on repeat.
When faced with this information, time is of absolute essence. The price of taking your time to doubt things and evaluating the veracity of the information means you're not acting to potentially prevent the immient death of yourself and your family.
In this scenario, you cannot expect people to do anything but take things at face value and seek immediate shelter.
This Twitter incident is a complete joke in comparison. While you should certainly be judicious if you're in any sort of significant communications role, I really don't care what's on USSTRATCOM's Twitter. I didn't even know they had a Twitter nor would I consider the information there more than a frequently updated newsletter. Give the employee a break, it's really not that big of a deal.