Bud Cline: Just because I know more about floor finishes that you do is no reason to act like you just sat on a tack. You need to grow up a bit.
Please don't use a floor "wax", which nowadays is called an "acrylic finish" on your Peel & Stick tiles unless you have a floor machine. That is, something that looks like one of the machines shown on this page:
Just to get the nomenclature out of the way, years ago they used Carnauba "wax" (obtained from the Brazilian Carnauba Palm tree nut) on hardwood floors. Then the S. C. Johnson Wax Co. came up with an acrylic coating for resiliant floors which contained no Carnauba wax at all, but which they foolishly called an "acrylic floor wax". So, the use of the word "wax" for any protective coating you put on a floor stuck, and now, people who don't know the difference use the term "floor wax" for the acrylic floor finishes that are available nowadays that contain no Carnauba wax at all. People that know the difference use the term "Finish" to refer to an acrylic floor "wax". Also, people that are in the business know the difference between a floor tile sealer and a floor tile finish.
Now, floor finishes are much softer than sealers. That's cuz a finish is made to be maintained by scrubbing off the surface layer that dirt gets embedded in underfoot, and then applying finish to replace the amount you scrubbed off. The standard maintenance procedure for floors with an acrylic finish on them is to periodically scrub the dirty surface layer off with a floor machine using an aggressive brown or blue pad, and then put several coats of new finish down, thereby restoring the floor to it's original clean and glossy appearance. You can keep doing that for decades, and one of the reasons vinyl composition tile floors are preferred by certain companies (like Safeway and many car dealerships) is that that you can keep vinyl composition tile floors looking new and shiny for decades (centuries, really) by just maintaining the floor finish on them.
If you don't have a floor machine, you simply can't scrub that dirty surface layer off. No human being has the strength to scrub that hard for that long as to scrub off the dirty surface of floor finish in a room. You use a floor machine to do that. It's just a big heavy electric motor that turns an abrasive nylon pad that scrubs that acrylic plastic off your floor.
If you don't have a floor machine to maintain your floor finish properly, it's a mistake to even put floor finish down. All you'll do is put down a soft coating that will become embedded with dirt in the traffic lanes much sooner, making for a floor you don't have the knowledge or equipment to maintain properly.
You could use a Magic Eraser to get rid of most of the dirt, but you could never get rid of all of it the way you could by scrubbing the dirty surface layer completely off. And, since the finish is softer, you'll be cleaning that floor finish much more often to get rid of the embedded dirt. Your only other option would be to strip all the finish off, and put new finish on, and you'd normally need to remove all the furniture in the room to do that.
Sealers dry to a much harder film than floor finishes because they're not intended to be removed or polished smooth. MUCH less dirt becomes embedded in the sealer because it is so much harder. So, it stays good looking much longer. I installed vinyl composition floor tiles in one sister's kitchen and my other sister's basement, and I used only sealer to give both floors a gloss. Both sisters are perfectly happy with their floors.
If you're wanting confirmation that what I'm telling you is correct, just look in your yellow pages phone directory under "Janitorial Equipment & Supplies" and talk to any of the people that cater to the janitorial service sector of the economy. They will all be knowledgeable about acrylic floor finishes and sealers made for vinyl composition tile floors.
If you're looking for a good floor sealer, contact any company that sells S. C. Johnson Wax products and buy a gallon of "Technique" or any company that sells "Buckeye" products and buy a gallon of "First Down". Those aren't the only good sealers on the market, but I've used them both on bathroom linoleum floors and know them both to dry to very hard and very protective (against stains) films.
If your tiles were made by Armstrong or Mannington, both companies will undoubtedly advise you to use their own products. Armstrong has a product called "New Beginnings" as a stripper and another product called "Shine Keeper" as a floor finish. When I was young and stupid enough to believe what the manufacturer told me, I would use their products TO MY OWN DETRIMENT. That's cuz both products are exhorbitantly priced, so people tend to use them sparingly, and that's just bad for proper maintenance. You would do MUCH better by buying ANY COMPANIES floor stripper and floor finish and using PLENTY of it to remove the finish from your floor completely and putting on MANY coats of finish so that you can "Scrub and Re-coat" for many years. If I recall, Armstrong's "Shine Keeper" costs about $30 for a half gallon. I use S. C. Johnson Wax "Carefree" on all my VC tile floors, and it costs about $30 per gallon. (I buy it by the 20 liter "enviropac".)
Truth is, that the flooring companies that make VC tiles, sheet vinyls and Peel & Stick tiles charge exhorbitant amounts for their products because they, and the retailer, know that the customer will buy them merely because they don't know what else to use. I'm telling you that you will maintain your floor much better by buying cheaper products from janitorial supply stores, and using MORE of them to both strip the finish off the floor more completely and provide a much thicker finish on the floor afterwards. Truth is, I never got a real high gloss shine on my VC tile floors when using Shinekeeper because I never used enough of the stuff to get a good shine. And, the reason why was cuz it cost $30 for a small bottle. When you're buying your finish or sealer for half that price, you use more, and you get a much thicker, more attractive and glossier film that provides more protection to your floor tiles.
I've had my fallings out with Armstrong, even though I have their VC tiles in all of my apartments. And, I can tell you that they screw their customers after the sale by charging double or triple the cost for the cleaners, strippers and finishes their customers need to maintain their sheet vinyl or VC tile floors, and I will debate Armstrong any day of the week on this point. Armstrong won't even admit that you need a floor machine to maintain the floor finish on a vinyl composition tile floor. I have little respect for them anymore. Maybe it's just their customer service department, none of whom have ever maintained vinyl composition tile floors. In my case, I own a 21 unit apartment block with VC tile floors in every apartment. You can see the condition of my VC tile floors on my web site at:
Keep in mind that every floor you see in the apartements available on my web site is at least 15 years old.