[The question was originally asked in an incorrect form, which is why there are two other answers that probably don't now appear to make sense. I'll give the answer to the question as it now appears, and then the answer to the question as it originally was.]
Suppose the heaviest-of-the-light is in row a and column b, and the lightest-of-the-heavy in row c and column d. Then the HOTL is lightest in its column and hence lighter than the egg in row c and column b; the LOTH is heaviest in its row and hence heavier than the egg in row c and column b; hence HOTL is lighter than LOTH. In other words: the blue egg is heavier than the red one.
[The question originally had some number of "lines" -- no distinction between rows and columns -- and took the heaviest and lightest eggs from each line. Here is what happens then:]
The heaviest-of-the-light can be heavier. Let's say the eggs have weights 1,2,...,200 in some units. Let one line have the eggs with weights 191,192,...,200, and arrange the others however you like. The lightest egg in that first line has weight 191 and is heavier than any egg in any other line.
The lightest-of-the-heavy can be heavier. Again, give the eggs weights 1,2,...,200. Now put eggs 181,182,...,200 into different lines, and again arrange the others however you like. The heaviest egg in each line has weight at least 181, which is heavier than any egg that isn't heaviest-in-its-line.
(The question originally had 10 lines of 20 instead of 20 lines of 10. I've adjusted the numbers above for the new version, but the same principle works no matter how many lines, and no matter how many eggs per line, we have.)