I installed self sticking vinyl squares and in no time they had separated and glue emerged to make the space between the squares turn hard black with dirt. What did I do wrong when installed to have the squares move and separate and what do I do now to apply to the spaces... grout?

What was under them?
If the subfloor is old wood planks, the normal expansion in warm weather will also expand the tiles and create spaces. You can't grout vinyl tile; there isn't enough depth for it to hold, and you have all that dirt in there now.
Personally I would put click flooring over it. Talk to the lumberyard or Home Depot about the different kinds. The fake wood grain ones are very cheap. Or use vinyl flooring by the yard.

What was under them?
If the subfloor is old wood planks, the normal expansion in warm weather will also expand the tiles and create spaces. You can't grout vinyl tile; there isn't enough depth for it to hold, and you have all that dirt in there now.
Personally I would put click flooring over it. Talk to the lumberyard or Home Depot about the different kinds. The fake wood grain ones are very cheap. Or use vinyl flooring by the yard.

Joypulv,
The subfloor is some kind of tile that was used in the 1960's. I always thought it was cement tiles. I have scraped out all the dirt between tiles and now there is a space for something. How about grout now or caulk? I need to do something inexpensive at this time because I also need new cabinets first before I install good flooring and I can not afford it at this time.