Your friend is completely right.
Correct boot fit is possible the single most important thing in your setup. You can ride a board that isn't completely ideal for you, a couple of cm longer or shorter isn't going to mean THAT much, and when it comes right down to it, bindings are more or less the same.
But a badly fitting boot will not only negatively impact your riding, it will also leave you with cold feet from bad circulation, blisters from pressure points (turning into open wounds if you keep riding) and constant pain while on the slopes.
You're right that 'any boot will do', as long as it fit. Don't cheapen out here, if you are on a tight budget it's possible to cheap out on a board or bindings, going for used or simply what is cheapest - but do this on boots are you're in for a very bad experience!
If you're lucky you might find a pair on sale that fits, went shopping with a girl I know that needed boots, and the boot we ended up finding that fit perfectly was a mere $85. But when I went shopping for new boots last season that only boots I could find that fitted was $350, including a 20% saving as it was the model from the year before.