I purchased a Jig Saw (mine is actually an old Black & Decker; the HF model is illustrative of how inexpensive they are), a few years ago, at an estate sale, but not for paint mixing. I paid $10 USD for it. Recently, I found videos on YouTube, on how I could purchase $2 clamps, at Harbor Freight, which would fit into the blade clamp on almost any Jig Saw…
The HF clamps were too thick to fit into the blade clamp on my Jig Saw. Still, I persevered, as the concept was too effective, to give up so easily. I visited my local DIY store, and came up with the idea of attaching the paint bottles to a regular, inexpensive blade, which would fit into my saw… It worked superbly.
In the interest of proof of concept, I used blue painter's tape, to secure an empty pill bottle to the blade, inserting the smaller paint bottles into this, and securing the locking cap in place. It mixes paint super-fast, super-easy. I do NOT recommend going full-throttle on the saw, however, as it may tear the tape off of the blade!
I wrapped the tape around the blade, and the bottle, twice. The paint bottles were the small, Reaper premium paints, as they were all I had which would fit inside the smaller pill bottle. The mixing was impressive, at moderate speed, for all of 30 seconds.
I highly recommend using some sort of container, to hold the paint bottles in place. The container should have a sealing cap, to contain the paint, should the bottle rupture (very unlikely). Blue painter's tape is very inexpensive, and easy to use. It is not excessively strong, and it is easy to peel off, when necessary. Duct tape is much stronger, and may suit your needs better.
If you really want to get crafty, you can drill a couple of holes in the steel blade, and attach the bottle with two, small bolts (I suggest round heads to avoid puncturing the paint bottles), and locking nuts. Get creative, knock it out of the park with 'over-kill', but please be safe, and careful. To paraphrase Miracle Max and his wife, Valerie, from The Princess Bride movie:
Max: "Have fun crafting your paint mixer, Boys!"
Valerie: "You think it will work?"
Max: "It will take a miracle…"
Together: "Good-bye, Boys!"
LOL! Cheers!