Most machines will have the legal verbiage somewhere, but in my several years of squishing, I've never stopped to read it. It's legal and I have better things to remember than the exact reasons in legalese.
Watch those 'shiny' pennies, though. Pre-1982 pennies were made entirely of copper - more recent ones have a zinc center, which is why you can sometimes see silvery streaks when you squish newer pennies. That silver color will turn to black over time, ruining the look of the squishie. Much better to shine up some older pennies if you like them shiny - a quick google will give you several easy methods. That way, you have solid copper coins and they'll stay nice. Personally, as long as they aren't a mess, I like the patina of the older coins.
And yes, it really is your penny that gets squished. There are some exceptions, mainly outside of the US in places with coins in different metals or different laws - I got one in Canada that was a copper blank to start. If it is a blank like that, the machine won't ask you to put in a penny since it isn't needed.
HTH! Google for more (and probably better) info
Oh, the original question..... I carry a small ziploc baggie with pennies and any quarters I have (most, but not all, gift shops will give you change if there isn't a machine around) and just put the squished pennies back into the same baggie. Since I use a backpack or put the ziploc in my pocket, I can carry quite a few without it seeming too heavy. When I use a shoulder bag or purse, the weight gets heavy quickly.