Resistence? RESISTENSE? The correct spelling is resistance.

You are right that negative resistance means the power flow is out of the material/device.

You are also correct to say that the measurement is wrong. I don't think you need professional opinion. The reason is quite simple. The probable case is calibration error. Also, the conductor measuring might be conducting current from an external source. In resistance measurement mode, a multimeter injects a small constant current say 0.1mA, then measure the voltage across the probe. If the measured voltage is 1.3mV, then the display shows 13 ohm. When the measured conductor is carrying current (caused by sources other than the multimeter itself), the voltage measured can be anything and can be negative.

A third explanation is similar to the second one. It is due to the measured object acting as a chemical cell. Set multimeter to millivolt scale. Put the two probes into a cup of water, I think you have done this. You'll see that it has voltage. If switch to resistance mode, you can't measure correctly. The readings obtained are very far from the readings you get from an analog ohmmeter, which use a higher test current. Same case for measuring the resistance from your left hand fingers to right hand fingers. Analog meter gives about 200kOhm, but digital gives rubbish (up and down over wide range). Now try measuring voltage on a piece of oxidized metal, if you touch the probes lightly, you are likely to get something more than 1mV. However, I've never success in getting negative resistance by this way.