Speakers? No. You need studio monitors. These are like speakers but they're flat, i.e. the frequency response is the same for all frequency ranges - bass, treble, mids. Normal speakers usually amp up the bass and treble. Which works for listening to music but if you're going to make your own, you won't get a real idea of how you sound.

You don't need a hardware mixer - you can just use a DAW (Digital Audio Workstation) on your computer and use that as a mixer.

Here's the physical setup I would recommend:

1. Durable, trustworthy mic (like Shure Sm58)
2. Connect the mic to an audio interface. Get something cheap from Focusrite
3. Connect the audio interface to your PC. Also connect the audio interface to your studio monitors
4. Load up a DAW on your PC. See this list for a selection of open source DAWs and mixing tools. If you're okay with paid solutions, consider FL Studio since its the cheapest around, though as a beatboxer, Ableton Live Lite should be good enough too.

Try to get second hand mics and monitors to save money.

This might not be the ideal beatboxing setup but it will form the foundation of a studio where you can do a lot more than just beatboxing.