So I am making my first foray into lace and I am at a loss. How do I not do a stitch and carry the yarn with me in the squares that are marked as black boxes?
I'm not entirely certain, but it sounds as though your "black box" might be an indication of where to put a yarn over - seeing the chart would make it much easier to decipher. In most lace charts this would be written as a circle in the box. This should be either preceded or followed by a decrease, though it's not absolutely essential that this occur adjacent to the yarn over itself. A yarn over creates a new stitch and makes a hole when the next row is knit. Eunny Jang has a good primer on knitting lace on her website. The link for part 3 of this (see below) has diagrams on the basic stitches that are used to make lace as well as standard chart notations.
"Hatred does not end by hatred; hatred ends by love. This is the eternal law." - Buddha
I think a black box would indicate where a stitch was but no longer is because it was decreased away in the previous row.
Again, we really need to see the chart. There is sort of a standardization coming about, but it seems that anyone writing a chart will use whatever symbols they want, and you have to decipher it from there. There normally is a legend somewhere telling you what all the symbols are, and how to do them.
From my understanding of most books, a solid black square usually indicates that there is nothing there. It is empty, no longer to be considered, a blank. Just ignore it.
Why do they use them then? As you will see, a row previous will have decreased some stitches, so there are fewer on this row. But to keep to a grid, they had to do something with those squares, so have filled them in solid, and they are to be thought of as nothing. Don't do anything with them!!
Now, some books will show a solid black square as a knit stitch! So you really need to read the instructions (I know, a radical move) and see what the author of the pattern had in mind.
If that still doesn't help, could you post part of the pattern, so we can see for ourselves? Or is from a book, commercial pattern, free site?