Does dual-booting an OS require a dual-core processor?

Your friend does not know what he's talking about.

Dual booting can be done on any computer regardless of CPU.
And dual or triple or quadruple booting will not slow your computer down, it only runs one OS at a time.

Dual booting is when you have multiple OS's installed on the same computer and you choose what OS to boot when you power it up.

The only way dual-booting affects the OS running right now is that the other OS takes some disk space. As long as you have enough disk space for both, you're done.

Your friend is incorrect. Dual booting has nothing to do with anything except the boot commands written to the hard drive.

It doesn't matter how many cores your CPU has, physical or virtual. It doesn't matter how many hard drives you have.

Technically, you can run more than one OS even on the same partition, though this usually only works with OSes of the same type, as in 2 versions of windows.

And if you dual-boot, your computer will not slow down. Installing another OS results in more hard drive space used up, but so long as you have sufficient capacity on your hard drive to fit the other OSes, you won't experience any slowdown related directly to the dual-boot configuration.

However, if you're talking of virtualization, which is where some software "pretends" it's another computer an an OS runs inside that software, yes, a multi-threaded, multi-core cpu is much better, and you'll experience a slowdown.

I recommend you do a little research on the OSes you want to run and why you want to run them, and then decide the best way to go about doing it.