If possible, use pennies minted before 1982.
In 1982, the Treasury started making pennies with a higher zinc (and lower copper) content. According to the Penny Collector Website:
Serious collectors will use pennies minted prior to 1982 because after 1982 the mint began to use a percentage of 99.2% zinc with a 0.8% copper - coating. In pre-1982 pennies that percentage was 95% copper, 5% zinc.
This higher zinc content means that, once squished, the silver color of the zinc will sometimes bleed through and often distracts from the design. Another problem that collectors have with zinc pennies is that the zinc often tarnishes and does not polish well. I have a penny that I pressed in the 1970s (from the Empire State Building). It has aged extremely well and looks beautiful. I have another penny that I pressed in the 1990s before I knew to use pre-1982 pennies (from the Golden Gate Bridge). It has tarnished badly and does not look very nice at all.
The switch from copper to zinc pennies occured during 1982 and so some 1982 pennies are copper and some are zinc. Many collectors avoid 1982 pennies altogether. But there are several ways to tell the difference. Some collectors swear that you can hear the difference if you drop the pennies on a hard surface. My method of choice is to weigh the pennies. Copper pennies are heavier. We have a small balance and Kyle loves to help me weigh my pennies and pull out the coppers.
While I do try to press only on copper for the above reasons, I am not that much of a snob. I would rather have the design on a zincer then not have the design at all. And I am extremely thankful to friends and family who bring me pennies from their trips, no matter what kind of penny it is pressed on.
When it comes to trading however, most collectors (including myself) seem to prefer coppers. Some folks will trade zincers, but only for other zincers and it is definitely a no-no to trade a zincer for a copper unless you agree ahead of time (this might be acceptable to some traders if the penny is retired but you still need to let the other person know).
Never guessed there was so much to know about penny squishing, did ya?