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You're not really using these dice as dice, but as markers.  It sounds like you won't be "rolling" these dice, and therefore expect an even outcome between landing on each of the 6 sides.

<h2>Paint them</h2>

The first solution is to get ordinary dice, then mark them yourself.  You can easily do this to 100s at a time.  Abut them all in a rectangle with the 6 side up, then hold them in place by clamping opposite sides of this rectangle.  Now run a paint roller across the top of the dice.

<h3>Don't spray</h3>

I see another suggestion says to use spray paint, but that's a bad idea.  The spray will get into cracks, and will be very hard to apply evenly.  Some might run down the edges.  The result will look messy and be more visible from the 4 adjacent sides.

You want the paint to be on the face only, not along the rounded edge to adjacent sides.  With the right kind of roller and thin layer of paint, it should look reasonably neat.

<h2>Use markers</h2>

This is actually an X-Y problem.  You want to distinguish between a piece being unclaimed or belonging to a particular player.  Address that directly.

You don't say how many of these pieces could be in play at any one time, but I'll assume the number is reasonably limited.  The game would include a dozen (whatever the max number is, plus a little) pieces of each color, or shape, or whatever you use to distinguish belonging to a particular player.  There would also be a dozen plain pieces.  When a player claims a piece, he swaps in his own.

<h3>Cut colored sheets</h3>

If the number could be large, make the player pieces simpler.  You are apparently OK with limiting the game to 5 players or less.  Get 6 different color plastic sheets and cut them up into squares.  They only need to be heavy and therefore thick enough to not blow away easily.  Paper squares wouldn't work, but even heavy cardboard might.  Think simple.

A few minutes with a paper cutter and 6 sheets of heavy colored cardboard should yield a large supply of playing pieces.

<h3>Patterns</h3>

For extra credit, you can print patterns on the sheets before you cut them up.  If you're making the squares 15 mm per edge, for example, then a repeating pattern 3-4 mm in pitch would work fine.  I'd leave the unclaimed markers unpatterned.  It shouldn't be hard to come up with 5 easily-distinguishable patterns, where color difference is only a bonus.

<h3>Printed overlays</h3>

Instead of printing on top of thick sheets, print ordinary paper.  With a color printer, you can print different colored patterns directly.  Use adhesive-backed sheets, and stick them to plastic or wood sheets.  5-10 mm thick would be good.  That's hefty enough to not easily blow away from someone walking by, and provides a little height to the pieces so they can be easily picked up with fingers.  Think of something like Scrabble tiles, although a little thicker would be better.

A good material might be a 3/8" sheet of particle board or "hardboard".  Those are easily cut in a grid pattern with a table saw.