Most listeners would think that the screech was an equipment malfunction, maybe due to an earthquake... and when the radio station went silent they'd probably switch to some other station or some other activity.
Depending on what the DJ says, how, and what the listener hears and feels some people might call the police, but most people will likely realize they don't know what happened, and surely someone is there, like a sound technician or someone if something happened... maybe the DJ was just tripping?
Nobody would start looking for ET unless they were gravely paranoid, but then they'd likely be out looking for ET already... (A blind hen finding an acorn?)
Someone at the station might get alarmed that the station went silent and start calling people. Maybe even head over to the radio station to check what's up, or ask someone there to do it for them.
The police will likely get involved if there is blood or signs of struggle in the studio.
If the DJ has just vanished without any sign of struggle or criminal activity the police would likely not start investigating until someone in the DJ's life makes a missing person report. If this doesn't happen, the police will likely consider him not missing...
A missing person report would probably not be investigated until 24 hours had passed since the disappearance. Especially if the DJ's car is in the parking lot at the radio studio and he cannot be suspected of having walked off into the wilderness. Maybe if it's winter, they'd be quicker.
Due to budgetary reasons the police will also declare the case a cold case and stop investigating it after a while. In that case, family members might hire a PI to keep looking. That PI-link also states that in the US 800,000 people are reported missing every year, so it would be another Tuesday at the precinct. Unless someone there knows the DJ...
Family members (that didn't listen to the radio) would probably be on the phone a while after he's supposed to have made contact or come home, and they would probably be out looking for him in places they think he might be an hour or two later. If they know about them, they might contact missing people (even before 24 hours have passed...)
If family members heard the radio show and what the DJ said they'd probably head over there immediately if they thought he was not joking or making a show.
As time passes the realization that the DJ is gone will settle and the reasoning behind why might differ from person to person from him having just up and left his life to him having been kidnapped or killed.
And since there will likely be no ransom demand, the police will also not investigate the disappearance as a kidnapping. Even if the DJ's wife is 100% sure he was kidnapped...
At around this time, it's likely people close to the DJ would start processing the situation in some variation of the stages of grief. This could take years, and some might never accept what happened.