Is it bad to mix different size tires on a car?

I tried to buy ONE tire to replace a cut tire on my Neon, and they refused to install it on the front.

I could have them install on the rear, leave and come right back, and have them rotate to the front, but direct install? No.

Their concern was new tires in the front could grip more than worn tires on the rear resulting in accidental oversteer which could cause a wreck.
I had the same encounter LOL, some stupid industry standard that really didnt relate to me, I understood what they were saying but it just didnt relate to MY driving abilities at all, I know how to handle MY own vehicle. Ended up buying two tires elsewhere and taking them elsewhere still to get mounted/balanced and put ON THE FRONT like I wanted. By the time that was done to my satisfaction I ended up saving $15 over the original quote LOL.