Rubbish. You have it backwards. Driving the LO of a balance mixer with a square wave results in the diodes or active devices operating like a switch. They're either on or off with nothing in between. If there were some added noise in between, such as if the rise and fall times on the square waves were slow, there might be some noise added during the transition by an active device. To prevent this, the FET and xsistor versions of the double balance mixer are driven hard to make sure they're totally on, or totally off.
However, if you were to drive the LO with a sine wave, there would be a longer time period when the active devices are operating as amplifiers instead of switches. That's where some noise MIGHT creep into the system. Even so, the most noise that can slither into the system is that produced by the LO itself, times whatever gain the active devices can deliver. With a 50 ohm load on the drain, there's not much gain there. Since the LO and its even harmonics are nulled out in the mixer, so will most of any added noise produced by the LO be nulled out. The final NF is also the average of these twice per LO cycle bursts of noise, thus again reducing the noise levels. However, if the LO is clean and low noise, none of these effects are going to be seen, and certainly not to the point of doubling the noise level (+6dB).