Communications sattelites in geosynchronous orbit don't carry enough propellant to lose enough velocity to achieve an orbit low enough to noticably increase their drag and accelerate their inevitable fall to earth. The stationkeeping thrusters only have enough energy to change their orbital by a few hundred km. It doesn't matter if you use the thrusters to increase the velocity and raise the orbit or decrease the velocity and lower the orbit. The energy available is only a tiny fraction, less than one percent, of the energy required to manoevre to either dip into the atmosphere and crash or achieve escape velocity and leave earth orbit.
The statement that implied the payload vehicle would continue to slowly drift further away from earth was mistaken. Increasing orbital height requires increasing velocity and that requires an ongoing energy input. The solar pressure from the sun averages out to nil over the long term and there's no appreciable atmospheric drag as there is in low earth orbit.
The statement preferring pushing dead satellites down rather than up ignores some practical considerations. If you push the thing downward, then everything ascending to geosynchronous orbit has to avoid it until it crashes, on a timescale marked in centuries. Unfortunately, there are a lot of upper stages left over from the launch vehicles that put these sattelites in orbit already there in highly eccentric orbits (highly eccentric implies their velocity at the peak is much lower than an object in circulr orbit at the same altitude). If the satellite is pushed downward its likely to eventually impact one or more of these eccentric orbits with a considerablly different velocity and produce a lot of small debris that can't be tracked and avoided.
So, the preference is to park the dead satellite up and out of the way. They also intentionally vent the remaining propellants, flatten the batteries, and so on to remove all onboard energy sources that might one day explode and spread debris.
There's nothing special about geosynchronous orbit other than that the obital period matches the planet's rotation, but since its kind of handy to not need tracking motors on satellite TV dishes, they're trying to minimize debris to keep the orbit useful for as long as possible. It won't last forvever, but maybe by then we won't need it anyway.
