No one makes it anymore?? That's amazing -- black-and-white tiles are my favorite kitchen look. Maybe somebody on eBay has a spare box somewhere?

Cork comes in very-nearly-white and very-nearly-black squares.

http://www.corkfloor.com/colours.html

Cork tiles my friend. Cool in summer, warm in winter, naturally antimicrobial, and noise dampening. Treat cork the same way you treat hardwood.

And so very kind on the feet and legs.

The tiles should run on the diagonal, not parallel to the walls.

As for health issues, I can't imagine what they would be unless you are thinking of something that isn't actually linoleum (the green flooring material). Obviously you shouldn't eat it, but linoleum itself is pretty safe; it gets used in LEED-certified buildings all the time. Sometimes people call vinyl flooring linoleum, which can be misleading.

And Marmoleum (a brand name for one company's linoleum) comes in white and black tiles.

Some linoleum was installed (in the past) with a glue that contained asbestos - and later 9" asbestos tiles were manufactured that looked like linoleum so there is sometimes concern about removing such tiles.
Marmoleum (just one brand of linoleum) does come in onyx and white marble. They both have a marbled texture to them but will show wear and dirt a lot less than a solid color will.
But since you're replacing the floor you could also consider doing a high contrast checker pattern with some color too.

Check out Armstrong Excelon composite tiles. It has a "flecked" pattern (not solid color), but it's 1/8" thick by 12" square.

http://www.armstrong.com/commflooringna/product_details_toolbox_magnify.jsp?item_id=47164

http://www.armstrong.com/commflooringna/product_details_toolbox_magnify.jsp?item_id=47165

Shocking that no one seems to make black and white linoleum. Good luck in your search, though; its such a great look.

I have black and white 12" linoleum tiles in my kitchen and bath! I ocassionally buy cheap replacements from Menards hardware store here in Michigan. Not sure if they have those stores where you live, but they totally sell the replacements you are looking for!

Try http://linoleumcity.com before you give up your search. They supply flooring for television and movie sets in LA and can usually get ANYTHING.

Send your question to retrorenovation.com. I bet Pam will be able to find a source for you in a jiffy!

If you're near SF, Discount Builders on Mission has them.

ikea has a linoleum product called Moran Modd that comes in black, gray and beige. We just installed the black in our kitchen.

http://www.ikea.com/us/en/catalog/products/20105336

The checkered stone tiles that linoleum took it's cue from didn't really come in snow white or jet black either. Have you seen the marmoleum tiles in person? I personally find the vinyl tiles that actually come in black and white a little stark. The real linoleum tiles look beautiful in natural light.

And in response to ebanfield lay of the tiles is a matter of taste. Tiles laid parallel to a wall can serve to lengthen a room or to compliment the way light falls through a window.
I think the room in the picture looks great.

They sell it at Home Depot. I just did my bedroom in this.

Considering cork flooring, it can come in virtually any color and style.

I just ordered 140 sf from a seller on ebay who ships free. I have used this product in several homes (kitchens, baths, mostly) over the last 20 years and still love it as much as ever. It is such a fresh look that can be accented so well with bright lively energetic colors like red, turquoise, yellow.