My outside airconditioning compressor comes on when I set the thermostat to "fan only". I was told by an hvac person the problem was probably the thermostat, so I replaced the thermostat, problem did not go away.

Probably a short between the green wire and yellow wire from thermostat to furnace or someone jumpered the "G" to "Y" at the furnace.

Disconnect wire from G terminal and try it again, this should tell you if you have a shorted wire with the Y terminal

I disconnected g wire, and tested... fan didn't come on, neither did compressor.
What's next? (thanks for the information by the way)

If you disconnected the G wire at the thermostat and the compressor does not come on when the thermostat is turned to "fan on" that make me think that you have a wire sorted to another wire some where between the furnace and the thermostat.

With the G wire disconnected does the condenser come on when turned to AC?

I never tested the a/c, didn't think about it! Was too concerned that the compressor was running. I put everything back the way it was, and discovered now, that the darn compressor was running at the same time the heater was running! I am guessing that is the new Hunter Thermostat... I am going to put the older thermostat back on and start over. For now, I pulled the circuit braker for the compressor, until I can find where the wiring problem is! Thank you!

Never buy hunter t-stats!!